The Quick Calculus Tutorial

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "The Quick Calculus Tutorial"

Transcription

1 The Quick Calculus Tutorial This text is a quick introuction into Calculus ieas an techniques. It is esigne to help you if you take the Calculus base course Physics 211 at the same time with Calculus I, but you o not yet have any Calculus backgroun. But we will assume that you have some algebra/trigonometry skills, of course you can refresh while you rea. This guie is accompanie by six short You-tube lectures going through the six sections (links see at the en of each section). Take a look at the vieos an rea the text below, it is not always quite the same things you will see. Do all the Exercises first yourself! Stop an think an calculate all Examples yourselves while you go along. Stop an rethink what you have learne all the time. There is also available the Calculus Concept Companion. These are notes from a course that introuces concepts with a ifferent time-line (having an eye on what s going on in your physic s class) in comparison to how the topics are covere in your section of Calculus I. Fell free to rea the companion an use its resources while you go through your Physics/Calculus courses. You can also rea the sections out of orer. 1

2 Lecture 1. What is the erivative? Calculus is the Mathematics of Change. A typical framework to introuce Calculus begins with the following question: How o we escribe the motion of a moving object? Three real numbers x(t), y(t), z(t) tell us an object s position at time t an thus tell us how the position is changing with time. Calculus begins with the question to escribe how fast the position is changing in time an to escribe the change in etail. This leas to the notion of velocity. For example x(1) = 2, y(1) = 2, z(1) = 1 means that at time t = 1 we reach the object from the origin by going two steps in the irection of the positive x-axis, then two steps in the irection of the positive y-axis an finally one step own in the irection of the negative z-axis. In this tutorial we will only consier things moving along a single axis. Often this may be a vertical motion like when you throw a ball vertically up an tell the height z(t) above groun at each point in time in some interval of time. There is a corresponing position functions z(t) (or sometimes x(t), y(t) for motions that are not vertical like a car moving along a straight line. For example imagine ropping a ball vertically own from a builing 50 feet above the groun. Suppose we place the origin of the z-axis at groun level. Then z(t) = 64 16t 2, 0 t 2 will be the height of the ball above groun for all times t in the time interval [0, 2]. When t = 2 then we have z(2) = = 0 an the ball hits the groun. This is an example of a function. We call t the inepenent variable an z the epenent variable in this case. 2

3 In Calculus books the notation is often ifferent. But this is just a matter of taste. (When we say y = 64 16x 2 for 0 x 2 an tell that x is time an y is the height above groun, we have given the same information, variable names on t matter but try to remember at all times what the variable names stan for.) Don t confuse the picture of the orbit of a moving object with the position graph. To the left you see the graph of the position graph of the object falling accoring to z(t) = 64 16t 2 uring the interval [0, 2]. Now the graph of the position function sloping own means that the object gets faster an faster on its way own, the spee is not constant. If the spee woul be constant the object woul fall equal istances in equal time intervals. If we set up a table showing the height z at times t we get t z an see that while the object only falls 4 feet within the first half secon it falls 28 feet within the fourth half secon. Let us consier another example. A car is moving along the positive x-axis. It starts from rest an accelerates within the first 10 secons. The following picture shows the graph of x(t). How fast is the car at t = 5? Imagine you take your feet from the peal for a short moment t at t = 5 an see how far you go. Note that if we stop accelerating we move with the constant velocity, which is the instantaneous spee that we ha at that moment. But how can we etermine this velocity? Suppose the position function is x(t) = 6t 2 3

4 If t is small the velocity will not change consierably in the interval 5 t 5 + t. So we can approximate the instantaneous velocity by the average velocity in this interval, which is for t = 0.1: v av = x t x(5.1) x(5) = Let us calculate this average velocity (note that you use the binomial formula) = 6 ( ) = = 6 ( ) If we replace 0.1 by t we get 6 (10 + t). Of course, when t gets smaller an smaller these number will get closer an closer to 6 10 = 60. This is the (instantaneous) velocity at t = 5. We write: v(5) = x t x = lim t=5 t 0 t Now we can o this calculation of the velocity for any time t. In this case we have the notation: v(t) = x t = lim x t 0 t You just have to be careful with the notation for x because it oes not specify for which t we consier the average velocity. x t = 6 (t + t)2 6t 2 t = 6 t2 + 2t t + ( t) 2 t 2 t = 12t + t When t get very small this quantity will be very close to 12t. Thus: v(t) = x t = 12t is the erivative of x(t). Note that v(5) = 12 5 = 60, which we calculate before. v(t) is the instantaneous velocity at time t. Now Calculus will be a collection of algebraic rules how to calculate erivatives without going through the above teious proceure of calculating a limit. But on t forget the interpretation of the numbers x (t) for a given function x(t): x (t) is the slope of the graph of x(t) for time t. In other wors: If x (t) > 0 the values of x(t) are increasing for time t: the velocity is positive. If x (t) = 0 the graph has a horizontal tangent, it usually changes from sloping up to own or vice 4

5 versa: the velocity is zero at this point. If x (t) < 0 then the values of x(t) are ecreasing at time time: the velocity is negative. Exercise: Consier x(t) = t 3. x t = t=1 x (1). Calculate x t for t = 1 an fin from this Watch the vieo Lecture 1 W hat is the Derivative? Lecture 2. How to take erivatives? In this section we will learn the basic rules of taking erivatives. I. Linearity: Imagine you are walking in a train moving along the x-axis so that x(t) is the coorinate of the center of the train, measure from some origin O. You measure how far to the right you are from the center of the train an call it y(t). Then your coorinate at time time is x(t)+y(t), which is the istance to the right from the origin O. What is your velocity with respect to groun: It obviously is the velocity of the train ae to your velocity with respect to the train. In Calculus language this means: x (x(t) + y(t)) = t t + y t. Obviously if you replace x(t) by kx(t) for a constant number k then also your velocity will be multiplie by k. Thus x (kx(t)) = k t t. 5

6 This shoul be clear from the following picture, noting that k x(t) k x(t + t) = k x. Note that if k < 0 you move in the other irection an the velocity also turns aroun. II. Power Rule: t tn = nt n 1, where n can be any real number. In particular t k = t (k 1) = t t0 = 0 for a constant k. (Get a feeling for the rule: First take the exponent as a factor own, then subtract one from the exponent.) Examples: The following examples are calculate using linearity an the power rule. 1. t (6t4 + 8t t + 4) = 6 4t t = 24t t So, for example the slope of x(t) = 6t 4 + 8t t + 4 (o you have an iea how the graph looks like?) for t = 1 is x t = = 60 t=1 2. t ( t + t 1/3 ) = t (t t 1 3 ) = 1 2 t ( 1 3 )t = 1 2 t t 4 3 = t t 4 3. Suppose you throw up a ball such that the height at time t is given by z(t) = 4 (t 2) 2 = 4 (t 2 4t + 4) = 4t t 2 = t(4 t). What is the velocity after 1 secon? (or what is the slope of the graph of the position function shown below). 6

7 We calculate: z t = t t (4t t2 ) = 4 2t, an so z t = 4 2 = 2. t=1 Does this answer look reasonable? Go from the point (1, 3) one unit to the right an 2 units up. This woul be the point you woul en up if gravity coul be cut off (which of course it can t) at t = 1. Note that z (2) = 0. This is the high point of the ball. There a few more important erivatives. t et = e t ; sin(t) = cos(t) ; t t ln(t) = 1 t cos(t) = sin(t) t Look at the graphs of the sine an cosine function an compare the slopes of sin(t) with the values of cos(t): 7

8 III. Prouct Rule an Quotient Rule: This tells how a prouct of two functions f(t) an g(t) is changing when we change the inepenent variable. f f (f(t)g(t)) = g(t) + f(t)g t t t The iea is to see the necessary quantities (fg) = ( f) g(t) + f(t) g + f g in the picture below. Here f(t) an g(t) enote the lengths of the two sies of a rectangle so that f(t) g(t) is the area. Note that for t small f g is very small so that: (fg) t = f t g(t) + f(t) g t By going over to the limit we get the prouct rule. Examples: t (t2 sin t) = ( t t2 ) sin t + t 2 ( t sin t) = 2t sin t + t2 cos t t (et sin t) = e t sin t + e t cos t = e t (sin t + cos t) Note that x(t) x(t) 1 1 = x(t) x(t) = 1 an the erivative of a constant is 0. So an application of the prouct rule gives 0 = t (x(t)x(t) 1 ) = x t x(t) 1 + x(t) t x(t) 1, 8

9 an thus by solving for t x(t) 1 we get x t t x(t) 1 = x(t) 2 Combining this with the prouct rule a simple calculation gives the quotient rule x t y = x y x y, y 2 where we have abbreviate x = x t an y = y t. Example: cos t sin t cos t t t tan t = sin t sin t t cos t = t cos 2 t cos t cos t sin t( sin t) = cos 2 t 1 = cos 2 t = sec2 t What is the erivative of e t? Let s look at the graphs of e t an e t : = cos2 t + sin 2 t cos 2 t Consier for example at the slopes of e t at t = 1. This is the negative of the slope of e t at t = 1. In fact in general we have that t e t = e t (Note that this means in particular: t e t t= 1 = e ( 1) = e 1 = t et t=1 ) 9

10 In general for each constant k: for example t e3t = 3e 3t. t ekt = ke kt, The above rule is actually a special case of the IV. Chain Rule: Consier a composition of two functions f(t) = x(u(t)) (check your Precalculus book if you forgot what this means!) What has this to o with e kt? Well recall that is is also written exp(kt), an we can consier this to be the composition of the function which multiplies by k an the exponential function, so u(t) = kt an x(u) = e u. Then the chain rule is: f t = x t = x u u t. The first term on the right han sie is more precisely x u, u(t) so you have to take the erivative an then substitute u(t) for the argument. Let s apply this to the example above: Then x u = u eu = e u an u t = t (kt) = k an the result follows. Examples: 10

11 1. Let x = e cos t. Then x(u) = e u an u(t) = cos t. Thus u eu = e u an x cos t = sin t x t = ecos t ( sin t) 2. Let x = cos(sin t). Then x(u) = cos(u) an u(t) = sin t. We get t cos(sin(t)) = u cos(u) sin t = sin(sin t) cos(t) sin t t 3. Suppose a particle moves along the z-axis an its position at time t is given by x(t) = e t2 +4t. Fin the velocity at time t = 1. This is x t t=1 We note that we have to fin the erivative of the composition of x(u) = e u with u(t) = t 2 + 4t. Note that for t = 1 we have u = = 3. Then x u = eu an u t = 2t + 4. Thus x t = x t=1 u u u=3 t = e 3 2 = 2e 3 t=1 Exercises: 1. Fin the erivatives x t : ˆ x = t 5 + t + 1 t (Hint: t = t 1/2 ) ˆ x = (t 2 + 6)e 3t ˆ x = e sin t cos(t) ˆ x = cos(cos(cos t)) ˆ x = ln cot t 2. Suppose an object moves along the y-axis an its position at time t is given by y(t) = t 2 cos(e t ). Fin the velocity at time t = ln( π 2 ). (Hint: You have to calculate t (t2 cos(e t )) t=ln( π ). The calculation of the 2 erivative requires the prouct rule, the chain rule, the power rule an the erivatives of e t an cos t). 11

12 Watch the vieo Lecture 2 How to calculate Derivatives? Lecture 3. How to calculate anti-erivatives? Given the function of velocities for a motion, how o we fin the position? Well certainly we nee to know aitionally an initial position, the velocity alone won t be able to tell us the position. But since the velocity is the erivative of the position, the position has to be an anti-erivative of the velocity. Note that such an anti-erivative can only be etermine up to a constant. We write v(t)t to enote the anti-erivative, which is a class of functions. Now what oes this mean? Let s say v(t) = t. Then v(t)t = tt = 1 2 t2 + C. Here C stans for an arbitrary constant. Note that t (1 2 t2 + C) = 1 2 t t2 + t C = 1 2 2t = t Let s say that for a motion along the x-axis we have given v(t) = 2t 2 an an initial position x(0) = 2. We first fin the anti-erivative of v(t): 2t 2 t = 2 3 t3 + C, so x(t) = 2 3 t3 + C for some constant C. position at time 0: an so C = 2 an the position is x(0) = C = 2 x(t) = 2 3 t This can be calculate from the But how i we know the anti-erivative of 2t 2. Well we can check that t (2 3 t3 + C) 2 3 3t2 = t 2. 12

13 In general taking anti-erivatives requires to turn aroun the rules we learne in Section 2. This is easy for some of the rules but har for others. It also requires you to get use to some conceptual unerstaning. If you take a erivative you actually perform an operation with input a function, let s say x, an output a function, then enote x t. The anti-erivative is an operation somehow turning this aroun. The anti-erivative has as input a function, let s say v(t), an as output a class of functions (a function plus all possible constants), then enote v(t)t. (We will explain the weir notation later on). Here are the basic rules: First linearity hols for integrals: (f(t) + g(t))t = f(t)t + g(t)t ; kf(t)t = k f(t)t, where k enotes a constant. t n 1 t = n + 1 tn+1 + C for n 1 (Get a feeling for the rule: First a 1 to the exponent, then take the reciprocal of the resulting number own as a factor.) Compare with the Power Rule. Note that the right han sie is not efine for n = 1. This is the exceptional rule: Examples: t t = ln t + C tt = t 1/2 t = 2 3 t3/2 + C 1 t + t5 t = ln t t6 + C Of course we have for constants k 0: e kt t = 1 k ekt + C sin(kt)t = 1 k cos(kt) + C ; cos(kt)t = 1 k sin(kt) + C 13

14 Note that you will never nee the above rules for k = 0 because e 0 = 1, sin(0) = 0 an cos(0) = 1. You will nee the special case of the Reverse Power Rule: kt = kt + C Examples: 1. Suppose that the velocity of a moving object is given by v(t) = 8 cos(2t) an the initial position is x(0) = 0. Fin the position function! We know that the position function is an anti-erivative for a specif value of the constant: 8 cos(2t)t = 8 cos(2t)t = 8 1 sin(2t) + C = 4 sin(2t) + C 2 Since 0 = x(0) = 4 sin(2 0) + C shows C = 0 we have x(t) = 4 sin(2t) 2. Suppose that x t = e t an x(1) = 1. Fin x(t)! We first fin the anti-erivative (k = 1): e t t = e t + C. Then x(1) = e 1 + C = 1 shows C = e an we get x(t) = e e t Check your answer: (1 t + 1 e e t ) = ( e t ) = e t an x(1) = e e 1 = 1 + e 1 e 1 = 1. Exercises: 1. Fin the anti-erivatives: ˆ t 2 + 2t 1/2 + t 1 t ˆ 2 cos t sin tt (Hint: t sin2 (t) = 2 cos t sin(t) from the chain rule.) 14

15 2. Consier the motion given by the velocity v(t) = y t = v 0 + at for a, v 0 constants an let y(0) = y 0. What is the position function y(t)? What is the meaning of a an v 0? Explain why t (v(t)) = t ( y t notation for the last function is 2 y t 2 erivative. 3. Fin 2 x t 2 for x(t) = t 4 + 3t 3 + 6t ) = a. The an is calle the secon orer Watch the vieo Lecture 3 How to calculate Anti erivatives? Lecture 4. The Funamental Theorem of Integral Calculus Suppose we are moving with constant velocity 50 miles/hr for two hours along the x-axis. How far i we get within those two hours? Of course, 100 miles. The velocity graph in this is a horizontal line at height 50. Note that we can interpret the istance covere in this case as the area of the rectangle boune by the lines x = 0, x = 50, t = 0 an t = 2. The units are in fact correct because 50 mi/hr 2 hr = 50miles. The number though can be interprete as an area. Of course this works in general: If we move with constant velocity k the position at time t will be kt, where we assume that the position at time 0 is 0. Again kt is the area of the rectangle with lengths t on the x-axis an height the velocity. Next assume that you are accelerating with constant acceleration a: v(t) = at along the x-axis. We know that if x(0) = 0. By taking the anti-erivative we know that: x(t) = 1 2 at2. How is this number relate with the graph of the velocity function? Is the number x(t) again the area uner the graph of x(t) > 0 an above the t-axis between 0 an t. In this case we the area is that of a triangle with base length t an height at so is 1 2 at2. By subtracting areas of rectangles we see that in this case the istance covere within the interval [t 1, t 2 ] is 1 2 a(t2 2 t2 1 ). 15

16 This is the motivation for the notation t 1 t 2 t 2 v(t)t = v(t)t = x(t) t 2 t 1 = x(t 2 ) x(t 1 ) t 1 This observation is true in general. The change in position is the net area between the graph of the velocity function an the t-axis. The integral a b f(t)t is calle the efinite integral of the function f(t) over the interval [a, b]. This is a number an not to be confuse with the anti-erivative (also calle inefinite integral), which is a class of functions. Net area means that we actually subtract the area above the graph an below the t-axis. This makes sense because when the velocity is negative we have to subtract from our position. This is inicate in the picture on the below. We have that 0 t v(t)t = A 1 A 2 = x(t) x(0) is the geometric interpretation of the anti-erivative. Now recall that the velocity is the erivative of the position an the position function is the anti-erivative. The general statement a b f(t)t = F (t 2 ) F (t 1 ) for a reasonably nice function f(t) an its anti-erivative F (t) is often calle the Funamental Theorem of Calculus. It tells the way how we calculate for example position from a given velocity. 16

17 A relate statement is that t t f(u)u = f(t) a Note that t a f(t)t is a specific anti-erivative of f(t), namely the one with f(a) = 0. But of course the erivative will not epen on which antierivative we pick. If we take the erivative of the integral we get back the function. Note that taking the integral of a erivative of a function gives back the function up to a constant. In fact, when we take the erivative we lose the initial conition an the integral won t know this information. In the picture below you get the visual representation of the Funamental Theorem assertions state above. In calculating efinite integrals we use some obvious rules, which are immeiate from the net area interpretation or the Funamental Theorem. For instance a c f(t)t = for any three numbers a, b, c. Also: b a a b f(t)t = f(t)t + a b b c f(t)t f(t)t Finally recall that evaluating a efinite integral is usually one by fining the anti-erivative first an then taking the ifference of the values of the anti-erivative at the two limits of the integral. Remark: You may woner about the strange symbol for integrals. This symbol is similar to the so calle Sigma symbol Σ use to abbreviate sums. 17

18 For example or 6 k=2 5 k=1 k = = 15 k 3 = = = 440 I believe you can guess how the symbol is use in general. This integral symbol is suppose to remin you of the fact that integrals are approximate by sums in the following way: Suppose we ivie an interval [a, b] into n small equal length sub-intervals of length t an pick values f(t i ) in the i-th interval. Then a b f(t)t n k=1 f(t k ) t Note that on the right han sie you are summing about areas of n rectangles with base lengths t an heights f(t k ). Examples: t 2 t = t3 = = 8 3. Note that this number is the area boune by the t-axis, the graph of t 2 an the line t = t 3 + sin πtt = 1 4 t4 1 1 π cos πt = π ( 1) ( π 0) = π t 5 + t 1/2 + e t 4t = t t3/2 + e t 4t = e What is 5 k=0 k 2? 5 k=0 k 2 = = = 55 18

19 Exercises: 1. Calculate the following efinite integrals an interpret in terms of net areas: ˆ π 0 sin(t)t ˆ t + t2 + t 3 t ˆ 1 0 e t t ˆ 1 1 f(t)t where f(t) is any o functions (which means f(t) = f( t)). 2. Suppose an object moves along the y-axis with velocity y t = cos(πt)+1 an initial position y(0) = 0. Determine y(1) by calculating a efinite integral. Interpret the resulting number as net area for the graph of a function. Watch the vieo Lecture 4 T he F unamental T heorem of Integral Calculus Lecture 5. Working with the tool box Newton s Law is usually state as F = ma where m is the mass an a = 2 x t 2 is the acceleration. Here we assume that F is a force acting on an object moving along the x-axis. Suppose that a force F (t) = cos t is the force an m = 1. How o we fin the position function x(t) of the particle if we know that x(0) = 1 an x (0) = 2. If we let v(t) enote the velocity along the x-axis then an so by taking the anti-erivative v t = cos t v(t) = sin t + C Since v(0) = x (0) = 2 we get C = 2 an so v(t) = 2 + sin t. 19

20 Then since x (t) = v(t) we get x(t) by taking the anti-erivative again: 2t cos t + C, for another constant C. Then from x(0) = 2 0 cos 0 + C = 1 + C + 1 we calculate C = 2 an thus x(t) = 2t cos t + 2. It is easy to check whether we have foun the correct solution. We fin by ifferentiating x (t) = 2 + sin t, 2 x = cos t, t t2 an x(0) = cos = 1, x (0) = 2 + sin 0 = 2. The following coorinate system shows the graphs of x(t), x t an 2 x t 2. Do you unerstan the relations between those graphs in terms of slope an net area? Exercise: Suppose an object with mass 2 kg moves along the y-axis. A force of F (t) = 2 e 2t Newtons acts on the object. We have y(0) = 1 an y t = 0. Fin the position function of the object an graph the function. t=0 Why is the object moving at all, taking into account that it s velocity at time 0 vanishes? Discuss the motion for large t. Watch the vieo Lecture 5 W orking with the tool box Lecture 6. Some Solutions for the Exercises Exercise 1.: We calculate x x(1 + t) x(1) = t t = t + t 2 = (1 + t)3 1 t = t + 3 t2 + t 3 1 t When t gets smaller an smaller, t 2 will be even smaller. Thus Exercises 2.1.: x t x = lim t=1 t 0 t = 3 20

21 ˆ x = t 5 + t + 1 t = t 5 + t 1/2 + t 1/2, an so by linearity an power rule: x t = 5t t 1/2 1 2 t 3/2 ˆ x = (t 2 + 6)e 3t, an by using the prouct rule, linearity, the power rule an the erivative of the exponential function (incluing a constant in the exponent): x t = ( t (t2 + 6)) e 3t + (t 2 + 6) t e3t = 2te 3t + (t 2 + 6) 3e 3t = e 3t (3t 2 + 2t + 18) ˆ x = e sin t cos t, an so by using prouct rule an chain rule we get: x t = ( t esin t ) cos t + e sin t t cos t = (e sin t cos t) cos t + e sin t ( sin t) = e sin t (cos 2 t sin t) ˆ x = cos(cos(cos(t)). We have to apply the chain rule twice: cos(cos(cos t)) = t u cos u u=cos(cos t) = sin(cos(cos t)) cos(cos t) t u cos u u=cos t t cos t = sin(cos(cos t)) ( sin(cos t)) ( sin t) = sin(cos(cos t)) sin(cos t) cos t ˆ x = ln(cot t). We calculate t cot t = cos t t sin t using the quotient rule: cos t sin t sin t cos t cos t = t sin t sin 2 = 1 t sin 2 t = csc2 t Exercise 2.2: y(t) = t 2 cos(e t ). We want to calculate y t t=ln π/2 21

22 We use prouct rule an chain rule to get y t = 2t cos(et ) + t 2 ( sin(e t ))e t = 2t cos(e t ) t 2 e t sin(e t ) an by evaluating at ln π/2 an using the ientity e ln u = u for all u we get y t = 2 ln π t=ln π/2 2 cos(π 2 ) (ln π π 2 )2 2 sin(π 2 ) = π 2 (ln π 2 ) Exercise 3.1: ˆ By linearity an the power rule for anti-erivatives (the power rule backwars) we get t 2 + 2t 1/2 + t 1 t = 1 3 t t3/2 + ln t + C = t t2/3 + ln t + C ˆ Since from the chain rule t sin2 t = 2 sin t cos t we know that the erivatives of sin 2 t is our integran an thus Exercise 3.2: Since v(t) = y t 2 cos t sin tt = sin 2 t + C we know that y(t) is an anti-erivative of v(t): vt = v 0 + att = v 0 t at2 + C The constant can be etermine by evaluating at t = 0: so the constant is y 0. So we have: v a 0 + C = C = y 0, y(t) = y 0 + v 0 t at2. The velocity is not constant but changing uniformly accoring to v t = a, where a is the constant acceleration. Also v(0) = v 0 is the initial velocity. This is the uniformly accelerate motion. 22

23 Exercise 3.3: From x = t 4 + 3t 3 + 6t we get an Exercise 4.1: x t = 4t3 + 9t t 2 x t 2 = t (x t ) = t (4t3 + 9t t) = 12t t + 12 ˆ π 0 sin tt = cos t π 0 = cos π + cos 0 = ( 1) + 1 = 2 ˆ t + t2 + t 3 t = t + t2 2 + t3 3 + t = = = ˆ 1 0 e t t = e t 1 0 = e 1 ( e 0 ) = 1 1 e ˆ The efinite integral is zero. This is because the integral is the net area, an counts area below the t-axis negative an the area above the t-axis positive. The two contributions will cancel by the symmetry (see graph below) 23

24 In the first three cases the graph of the function is above the t-axis over the corresponing interval. So the efinite integral is the blue area. In the last case the contributions on the negative axis cancel with those on the positive axis. Exercise 4.2: Let v(t) = cos(πt) + 1. For a general number τ the efinite integral y(τ) = 0 τ v(t)t is the anti-erivative with y(0) = 0. This is because 0 0 g(t)t = 0 for every function g(t). Thus y(1) = 0 1 cos(πt) + 1t is the efinite integral calculating y(1). Using the Funamental Theorem we get y(1) = 1 1 π sin(πt) + t = 1 sin(π) + 1 (sin(0) + 0) = 1. π Compare this with the shae area for the graph below: 0 Exercise 5.1: From Newton s law m = 2 an F = m 2 y t 2 we get 2 y t 2 = e 2t. Recall that we have also given y(0) = 1 an y t = 0. t=0 anti-erivative we get 2 y t t = t e 2t + C By taking the 24

25 an using the Funamental theorem an y t = 0 we get 1 t=0 4 + C = 0 an thus C = 1 4. Thus y t = t e 2t 1 4 Again we take the anti-erivative an get y t t = t e 2t 1 4 = t e 2t t 4 + C From y(0) = 1 we get C = 1 an thus C = 7 8. This gives y(t) = 7 8 t 4 + t e 2t You may want to check whether this y(t) satisfies the conitions: y(0) = e 2 0 = = 1 an by taking erivatives y t = t e 2t, 2 y t 2 = e 2t The object is moving because even though the velocity at t = 0 vanishes the acceleration oes not. This is like ropping a ball from rest, it starts with velocity zero but the acceleration ue to gravity makes the object move. Below you the the graph of y(t). Can you relate the graph to the given force? For large t we have F 2 is constant an the motion approaches a uniformly accelerate motion with position graph a parabola (what is the formula for the parabola?) Watch the vieo Lecture 6 Some Solutions f or the Exercises 25

MATH 125: LAST LECTURE

MATH 125: LAST LECTURE MATH 5: LAST LECTURE FALL 9. Differential Equations A ifferential equation is an equation involving an unknown function an it s erivatives. To solve a ifferential equation means to fin a function that

More information

Lecture 13: Differentiation Derivatives of Trigonometric Functions

Lecture 13: Differentiation Derivatives of Trigonometric Functions Lecture 13: Differentiation Derivatives of Trigonometric Functions Derivatives of the Basic Trigonometric Functions Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the

More information

CHAPTER 5 : CALCULUS

CHAPTER 5 : CALCULUS Dr Roger Ni (Queen Mary, University of Lonon) - 5. CHAPTER 5 : CALCULUS Differentiation Introuction to Differentiation Calculus is a branch of mathematics which concerns itself with change. Irrespective

More information

arcsine (inverse sine) function

arcsine (inverse sine) function Inverse Trigonometric Functions c 00 Donal Kreier an Dwight Lahr We will introuce inverse functions for the sine, cosine, an tangent. In efining them, we will point out the issues that must be consiere

More information

Here the units used are radians and sin x = sin(x radians). Recall that sin x and cos x are defined and continuous everywhere and

Here the units used are radians and sin x = sin(x radians). Recall that sin x and cos x are defined and continuous everywhere and Lecture 9 : Derivatives of Trigonometric Functions (Please review Trigonometry uner Algebra/Precalculus Review on the class webpage.) In this section we will look at the erivatives of the trigonometric

More information

M147 Practice Problems for Exam 2

M147 Practice Problems for Exam 2 M47 Practice Problems for Exam Exam will cover sections 4., 4.4, 4.5, 4.6, 4.7, 4.8, 5., an 5.. Calculators will not be allowe on the exam. The first ten problems on the exam will be multiple choice. Work

More information

Answers to the Practice Problems for Test 2

Answers to the Practice Problems for Test 2 Answers to the Practice Problems for Test 2 Davi Murphy. Fin f (x) if it is known that x [f(2x)] = x2. By the chain rule, x [f(2x)] = f (2x) 2, so 2f (2x) = x 2. Hence f (2x) = x 2 /2, but the lefthan

More information

Introduction to Integration Part 1: Anti-Differentiation

Introduction to Integration Part 1: Anti-Differentiation Mathematics Learning Centre Introuction to Integration Part : Anti-Differentiation Mary Barnes c 999 University of Syney Contents For Reference. Table of erivatives......2 New notation.... 2 Introuction

More information

Lecture 17: Implicit differentiation

Lecture 17: Implicit differentiation Lecture 7: Implicit ifferentiation Nathan Pflueger 8 October 203 Introuction Toay we iscuss a technique calle implicit ifferentiation, which provies a quicker an easier way to compute many erivatives we

More information

Inverse Trig Functions

Inverse Trig Functions Inverse Trig Functions c A Math Support Center Capsule February, 009 Introuction Just as trig functions arise in many applications, so o the inverse trig functions. What may be most surprising is that

More information

Lecture L25-3D Rigid Body Kinematics

Lecture L25-3D Rigid Body Kinematics J. Peraire, S. Winall 16.07 Dynamics Fall 2008 Version 2.0 Lecture L25-3D Rigi Boy Kinematics In this lecture, we consier the motion of a 3D rigi boy. We shall see that in the general three-imensional

More information

Math 230.01, Fall 2012: HW 1 Solutions

Math 230.01, Fall 2012: HW 1 Solutions Math 3., Fall : HW Solutions Problem (p.9 #). Suppose a wor is picke at ranom from this sentence. Fin: a) the chance the wor has at least letters; SOLUTION: All wors are equally likely to be chosen. The

More information

Exponential Functions: Differentiation and Integration. The Natural Exponential Function

Exponential Functions: Differentiation and Integration. The Natural Exponential Function 46_54.q //4 :59 PM Page 5 5 CHAPTER 5 Logarithmic, Eponential, an Other Transcenental Functions Section 5.4 f () = e f() = ln The inverse function of the natural logarithmic function is the natural eponential

More information

f(x) = a x, h(5) = ( 1) 5 1 = 2 2 1

f(x) = a x, h(5) = ( 1) 5 1 = 2 2 1 Exponential Functions an their Derivatives Exponential functions are functions of the form f(x) = a x, where a is a positive constant referre to as the base. The functions f(x) = x, g(x) = e x, an h(x)

More information

The Inverse Trigonometric Functions

The Inverse Trigonometric Functions The Inverse Trigonometric Functions These notes amplify on the book s treatment of inverse trigonometric functions an supply some neee practice problems. Please see pages 543 544 for the graphs of sin

More information

Lagrangian and Hamiltonian Mechanics

Lagrangian and Hamiltonian Mechanics Lagrangian an Hamiltonian Mechanics D.G. Simpson, Ph.D. Department of Physical Sciences an Engineering Prince George s Community College December 5, 007 Introuction In this course we have been stuying

More information

Calculating Viscous Flow: Velocity Profiles in Rivers and Pipes

Calculating Viscous Flow: Velocity Profiles in Rivers and Pipes previous inex next Calculating Viscous Flow: Velocity Profiles in Rivers an Pipes Michael Fowler, UVa 9/8/1 Introuction In this lecture, we ll erive the velocity istribution for two examples of laminar

More information

Section 3.3. Differentiation of Polynomials and Rational Functions. Difference Equations to Differential Equations

Section 3.3. Differentiation of Polynomials and Rational Functions. Difference Equations to Differential Equations Difference Equations to Differential Equations Section 3.3 Differentiation of Polynomials an Rational Functions In tis section we begin te task of iscovering rules for ifferentiating various classes of

More information

Given three vectors A, B, andc. We list three products with formula (A B) C = B(A C) A(B C); A (B C) =B(A C) C(A B);

Given three vectors A, B, andc. We list three products with formula (A B) C = B(A C) A(B C); A (B C) =B(A C) C(A B); 1.1.4. Prouct of three vectors. Given three vectors A, B, anc. We list three proucts with formula (A B) C = B(A C) A(B C); A (B C) =B(A C) C(A B); a 1 a 2 a 3 (A B) C = b 1 b 2 b 3 c 1 c 2 c 3 where the

More information

SOME NOTES ON THE HYPERBOLIC TRIG FUNCTIONS SINH AND COSH

SOME NOTES ON THE HYPERBOLIC TRIG FUNCTIONS SINH AND COSH SOME NOTES ON THE HYPERBOLIC TRIG FUNCTIONS SINH AND COSH Basic Definitions In homework set # one of the questions involves basic unerstaning of the hyperbolic functions sinh an cosh. We will use this

More information

Solutions to modified 2 nd Midterm

Solutions to modified 2 nd Midterm Math 125 Solutions to moifie 2 n Miterm 1. For each of the functions f(x) given below, fin f (x)). (a) 4 points f(x) = x 5 + 5x 4 + 4x 2 + 9 Solution: f (x) = 5x 4 + 20x 3 + 8x (b) 4 points f(x) = x 8

More information

Rules for Finding Derivatives

Rules for Finding Derivatives 3 Rules for Fining Derivatives It is teious to compute a limit every time we nee to know the erivative of a function. Fortunately, we can evelop a small collection of examples an rules that allow us to

More information

Example Optimization Problems selected from Section 4.7

Example Optimization Problems selected from Section 4.7 Example Optimization Problems selecte from Section 4.7 19) We are aske to fin the points ( X, Y ) on the ellipse 4x 2 + y 2 = 4 that are farthest away from the point ( 1, 0 ) ; as it happens, this point

More information

Trigonometry CheatSheet

Trigonometry CheatSheet Trigonometry CheatSheet 1 How to use this ocument This ocument is not meant to be a list of formulas to be learne by heart. The first few formulas are very basic (they escen from the efinition an/or Pythagoras

More information

Inverse Trig Functions

Inverse Trig Functions Inverse Trig Functions Previously in Math 30 DEFINITION 24 A function g is the inverse of the function f if g( f ()) = for all in the omain of f 2 f (g()) = for all in the omain of g In this situation

More information

Topic 2: The Trigonometric Ratios Finding Sides

Topic 2: The Trigonometric Ratios Finding Sides Topic 2: The Trigonometric Ratios Fining Sies Labelling sies To use the Trigonometric Ratios, commonly calle the Trig Ratios, it is important to learn how to label the right angle triangle. The hypotenuse

More information

Learning Objectives for Math 165

Learning Objectives for Math 165 Learning Objectives for Math 165 Chapter 2 Limits Section 2.1: Average Rate of Change. State the definition of average rate of change Describe what the rate of change does and does not tell us in a given

More information

Solutions to Examples from Related Rates Notes. ds 2 mm/s. da when s 100 mm

Solutions to Examples from Related Rates Notes. ds 2 mm/s. da when s 100 mm Solutions to Examples from Relate Rates Notes 1. A square metal plate is place in a furnace. The quick temperature change causes the metal plate to expan so that its surface area increases an its thickness

More information

Fluid Pressure and Fluid Force

Fluid Pressure and Fluid Force 0_0707.q //0 : PM Page 07 SECTION 7.7 Section 7.7 Flui Pressure an Flui Force 07 Flui Pressure an Flui Force Fin flui pressure an flui force. Flui Pressure an Flui Force Swimmers know that the eeper an

More information

Proof of the Power Rule for Positive Integer Powers

Proof of the Power Rule for Positive Integer Powers Te Power Rule A function of te form f (x) = x r, were r is any real number, is a power function. From our previous work we know tat x x 2 x x x x 3 3 x x In te first two cases, te power r is a positive

More information

Calculus Refresher, version 2008.4. c 1997-2008, Paul Garrett, garrett@math.umn.edu http://www.math.umn.edu/ garrett/

Calculus Refresher, version 2008.4. c 1997-2008, Paul Garrett, garrett@math.umn.edu http://www.math.umn.edu/ garrett/ Calculus Refresher, version 2008.4 c 997-2008, Paul Garrett, garrett@math.umn.eu http://www.math.umn.eu/ garrett/ Contents () Introuction (2) Inequalities (3) Domain of functions (4) Lines (an other items

More information

Hyperbolic functions (CheatSheet)

Hyperbolic functions (CheatSheet) Hyperbolic functions (CheatSheet) 1 Intro For historical reasons hyperbolic functions have little or no room at all in the syllabus of a calculus course, but as a matter of fact they have the same ignity

More information

Notes on tangents to parabolas

Notes on tangents to parabolas Notes on tangents to parabolas (These are notes for a talk I gave on 2007 March 30.) The point of this talk is not to publicize new results. The most recent material in it is the concept of Bézier curves,

More information

Inverse Trig Functions

Inverse Trig Functions Inverse Trig Functions Trig functions are not one-to-one, so we can not formally get an inverse. To efine the notion of inverse trig functions we restrict the omains to obtain one-to-one functions: [ Restrict

More information

Math 229 Lecture Notes: Product and Quotient Rules Professor Richard Blecksmith richard@math.niu.edu

Math 229 Lecture Notes: Product and Quotient Rules Professor Richard Blecksmith richard@math.niu.edu Mat 229 Lecture Notes: Prouct an Quotient Rules Professor Ricar Blecksmit ricar@mat.niu.eu 1. Time Out for Notation Upate It is awkwar to say te erivative of x n is nx n 1 Using te prime notation for erivatives,

More information

Week 13 Trigonometric Form of Complex Numbers

Week 13 Trigonometric Form of Complex Numbers Week Trigonometric Form of Complex Numbers Overview In this week of the course, which is the last week if you are not going to take calculus, we will look at how Trigonometry can sometimes help in working

More information

4. Important theorems in quantum mechanics

4. Important theorems in quantum mechanics TFY4215 Kjemisk fysikk og kvantemekanikk - Tillegg 4 1 TILLEGG 4 4. Important theorems in quantum mechanics Before attacking three-imensional potentials in the next chapter, we shall in chapter 4 of this

More information

20. Product rule, Quotient rule

20. Product rule, Quotient rule 20. Prouct rule, 20.1. Prouct rule Prouct rule, Prouct rule We have seen that the erivative of a sum is the sum of the erivatives: [f(x) + g(x)] = x x [f(x)] + x [(g(x)]. One might expect from this that

More information

Definition of Vertical Asymptote The line x = a is called a vertical asymptote of f (x) if at least one of the following is true: f (x) =

Definition of Vertical Asymptote The line x = a is called a vertical asymptote of f (x) if at least one of the following is true: f (x) = Vertical Asymptotes Definition of Vertical Asymptote The line x = a is called a vertical asymptote of f (x) if at least one of the following is true: lim f (x) = x a lim f (x) = lim x a lim f (x) = x a

More information

Taylor Polynomials and Taylor Series Math 126

Taylor Polynomials and Taylor Series Math 126 Taylor Polynomials and Taylor Series Math 26 In many problems in science and engineering we have a function f(x) which is too complicated to answer the questions we d like to ask. In this chapter, we will

More information

MOOCULUS. massive open online calculus C A L C U L U S T H I S D O C U M E N T W A S T Y P E S E T O N A P R I L 1 0,

MOOCULUS. massive open online calculus C A L C U L U S T H I S D O C U M E N T W A S T Y P E S E T O N A P R I L 1 0, MOOCULUS massive open online calculus C A L C U L U S T H I S D O C U M E N T W A S T Y P E S E T O N A P R I L 0, 2 0 4. 2 Copyright c 204 Jim Fowler an Bart Snapp This work is license uner the Creative

More information

Section 3.1 Worksheet NAME. f(x + h) f(x)

Section 3.1 Worksheet NAME. f(x + h) f(x) MATH 1170 Section 3.1 Worksheet NAME Recall that we have efine the erivative of f to be f (x) = lim h 0 f(x + h) f(x) h Recall also that the erivative of a function, f (x), is the slope f s tangent line

More information

6-4 : Learn to find the area and circumference of circles. Area and Circumference of Circles (including word problems)

6-4 : Learn to find the area and circumference of circles. Area and Circumference of Circles (including word problems) Circles 6-4 : Learn to fin the area an circumference of circles. Area an Circumference of Circles (incluing wor problems) 8-3 Learn to fin the Circumference of a circle. 8-6 Learn to fin the area of circles.

More information

Worksheet 1. What You Need to Know About Motion Along the x-axis (Part 1)

Worksheet 1. What You Need to Know About Motion Along the x-axis (Part 1) Worksheet 1. What You Need to Know About Motion Along the x-axis (Part 1) In discussing motion, there are three closely related concepts that you need to keep straight. These are: If x(t) represents the

More information

2012-2013 Enhanced Instructional Transition Guide Mathematics Algebra I Unit 08

2012-2013 Enhanced Instructional Transition Guide Mathematics Algebra I Unit 08 01-013 Enhance Instructional Transition Guie Unit 08: Exponents an Polynomial Operations (18 ays) Possible Lesson 01 (4 ays) Possible Lesson 0 (7 ays) Possible Lesson 03 (7 ays) POSSIBLE LESSON 0 (7 ays)

More information

Sections 3.1/3.2: Introducing the Derivative/Rules of Differentiation

Sections 3.1/3.2: Introducing the Derivative/Rules of Differentiation Sections 3.1/3.2: Introucing te Derivative/Rules of Differentiation 1 Tangent Line Before looking at te erivative, refer back to Section 2.1, looking at average velocity an instantaneous velocity. Here

More information

APPLICATION OF CALCULUS IN COMMERCE AND ECONOMICS

APPLICATION OF CALCULUS IN COMMERCE AND ECONOMICS Application of Calculus in Commerce an Economics 41 APPLICATION OF CALCULUS IN COMMERCE AND ECONOMICS æ We have learnt in calculus that when 'y' is a function of '', the erivative of y w.r.to i.e. y ö

More information

10.2 Systems of Linear Equations: Matrices

10.2 Systems of Linear Equations: Matrices SECTION 0.2 Systems of Linear Equations: Matrices 7 0.2 Systems of Linear Equations: Matrices OBJECTIVES Write the Augmente Matrix of a System of Linear Equations 2 Write the System from the Augmente Matrix

More information

Mathematics Review for Economists

Mathematics Review for Economists Mathematics Review for Economists by John E. Floy University of Toronto May 9, 2013 This ocument presents a review of very basic mathematics for use by stuents who plan to stuy economics in grauate school

More information

Modelling and Calculus

Modelling and Calculus Moelling an Calculus By Collin Phillips This material has been evelope as a joint project between the Mathematics Learning Centre MLC an the Learning Centre LC at the University of Syney. With literary

More information

Differential Calculus: Differentiation (First Principles, Rules) and Sketching Graphs (Grade 12)

Differential Calculus: Differentiation (First Principles, Rules) and Sketching Graphs (Grade 12) OpenStax-CNX moule: m39313 1 Differential Calculus: Differentiation (First Principles, Rules) an Sketcing Graps (Grae 12) Free Hig Scool Science Texts Project Tis work is prouce by OpenStax-CNX an license

More information

correct-choice plot f(x) and draw an approximate tangent line at x = a and use geometry to estimate its slope comment The choices were:

correct-choice plot f(x) and draw an approximate tangent line at x = a and use geometry to estimate its slope comment The choices were: Topic 1 2.1 mode MultipleSelection text How can we approximate the slope of the tangent line to f(x) at a point x = a? This is a Multiple selection question, so you need to check all of the answers that

More information

Measures of distance between samples: Euclidean

Measures of distance between samples: Euclidean 4- Chapter 4 Measures of istance between samples: Eucliean We will be talking a lot about istances in this book. The concept of istance between two samples or between two variables is funamental in multivariate

More information

Mathematics. Circles. hsn.uk.net. Higher. Contents. Circles 119 HSN22400

Mathematics. Circles. hsn.uk.net. Higher. Contents. Circles 119 HSN22400 hsn.uk.net Higher Mathematics UNIT OUTCOME 4 Circles Contents Circles 119 1 Representing a Circle 119 Testing a Point 10 3 The General Equation of a Circle 10 4 Intersection of a Line an a Circle 1 5 Tangents

More information

Chapter 4 One Dimensional Kinematics

Chapter 4 One Dimensional Kinematics Chapter 4 One Dimensional Kinematics 41 Introduction 1 4 Position, Time Interval, Displacement 41 Position 4 Time Interval 43 Displacement 43 Velocity 3 431 Average Velocity 3 433 Instantaneous Velocity

More information

Scalar : Vector : Equal vectors : Negative vectors : Proper vector : Null Vector (Zero Vector): Parallel vectors : Antiparallel vectors :

Scalar : Vector : Equal vectors : Negative vectors : Proper vector : Null Vector (Zero Vector): Parallel vectors : Antiparallel vectors : ELEMENTS OF VECTOS 1 Scalar : physical quantity having only magnitue but not associate with any irection is calle a scalar eg: time, mass, istance, spee, work, energy, power, pressure, temperature, electric

More information

How to Avoid the Inverse Secant (and Even the Secant Itself)

How to Avoid the Inverse Secant (and Even the Secant Itself) How to Avoi the Inverse Secant (an Even the Secant Itself) S A Fulling Stephen A Fulling (fulling@mathtamue) is Professor of Mathematics an of Physics at Teas A&M University (College Station, TX 7783)

More information

Angles and Quadrants. Angle Relationships and Degree Measurement. Chapter 7: Trigonometry

Angles and Quadrants. Angle Relationships and Degree Measurement. Chapter 7: Trigonometry Chapter 7: Trigonometry Trigonometry is the study of angles and how they can be used as a means of indirect measurement, that is, the measurement of a distance where it is not practical or even possible

More information

Homework 8. problems: 10.40, 10.73, 11.55, 12.43

Homework 8. problems: 10.40, 10.73, 11.55, 12.43 Hoework 8 probles: 0.0, 0.7,.55,. Proble 0.0 A block of ass kg an a block of ass 6 kg are connecte by a assless strint over a pulley in the shape of a soli isk having raius R0.5 an ass M0 kg. These blocks

More information

f(x) = undefined otherwise We have Domain(f) = [ π, π ] and Range(f) = [ 1, 1].

f(x) = undefined otherwise We have Domain(f) = [ π, π ] and Range(f) = [ 1, 1]. Lecture 6 : Inverse Trigonometric Functions Inverse Sine Function (arcsin = sin ) The trigonometric function sin is not one-to-one functions, hence in orer to create an inverse, we must restrict its omain.

More information

Exploration 7-2a: Differential Equation for Compound Interest

Exploration 7-2a: Differential Equation for Compound Interest Eploration 7-2a: Differential Equation for Compoun Interest Objective: Write an solve a ifferential equation for the amount of mone in a savings account as a function of time. When mone is left in a savings

More information

1 Derivatives of Piecewise Defined Functions

1 Derivatives of Piecewise Defined Functions MATH 1010E University Matematics Lecture Notes (week 4) Martin Li 1 Derivatives of Piecewise Define Functions For piecewise efine functions, we often ave to be very careful in computing te erivatives.

More information

1. the acceleration of the body decreases by. 2. the acceleration of the body increases by. 3. the body falls 9.8 m during each second.

1. the acceleration of the body decreases by. 2. the acceleration of the body increases by. 3. the body falls 9.8 m during each second. Answer, Key Homework 3 Davi McIntyre 45123 Mar 25, 2004 1 This print-out shoul have 21 questions. Multiple-choice questions may continue on the next column or pae fin all choices before makin your selection.

More information

2.2. Instantaneous Velocity

2.2. Instantaneous Velocity 2.2. Instantaneous Velocity toc Assuming that your are not familiar with the technical aspects of this section, when you think about it, your knowledge of velocity is limited. In terms of your own mathematical

More information

Course outline, MA 113, Spring 2014 Part A, Functions and limits. 1.1 1.2 Functions, domain and ranges, A1.1-1.2-Review (9 problems)

Course outline, MA 113, Spring 2014 Part A, Functions and limits. 1.1 1.2 Functions, domain and ranges, A1.1-1.2-Review (9 problems) Course outline, MA 113, Spring 2014 Part A, Functions and limits 1.1 1.2 Functions, domain and ranges, A1.1-1.2-Review (9 problems) Functions, domain and range Domain and range of rational and algebraic

More information

Mannheim curves in the three-dimensional sphere

Mannheim curves in the three-dimensional sphere Mannheim curves in the three-imensional sphere anju Kahraman, Mehmet Öner Manisa Celal Bayar University, Faculty of Arts an Sciences, Mathematics Department, Muraiye Campus, 5, Muraiye, Manisa, urkey.

More information

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 14 10/27/2008 MOMENT GENERATING FUNCTIONS

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 14 10/27/2008 MOMENT GENERATING FUNCTIONS MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 14 10/27/2008 MOMENT GENERATING FUNCTIONS Contents 1. Moment generating functions 2. Sum of a ranom number of ranom variables 3. Transforms

More information

1 of 7 9/5/2009 6:12 PM

1 of 7 9/5/2009 6:12 PM 1 of 7 9/5/2009 6:12 PM Chapter 2 Homework Due: 9:00am on Tuesday, September 8, 2009 Note: To understand how points are awarded, read your instructor's Grading Policy. [Return to Standard Assignment View]

More information

18.01 Single Variable Calculus Fall 2006

18.01 Single Variable Calculus Fall 2006 MIT OpenCourseWare http://ocw.mit.edu 8.0 Single Variable Calculus Fall 2006 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Unit : Derivatives A. What

More information

5.2 Unit Circle: Sine and Cosine Functions

5.2 Unit Circle: Sine and Cosine Functions Chapter 5 Trigonometric Functions 75 5. Unit Circle: Sine and Cosine Functions In this section, you will: Learning Objectives 5..1 Find function values for the sine and cosine of 0 or π 6, 45 or π 4 and

More information

Finding Antiderivatives and Evaluating Integrals

Finding Antiderivatives and Evaluating Integrals Chapter 5 Finding Antiderivatives and Evaluating Integrals 5. Constructing Accurate Graphs of Antiderivatives Motivating Questions In this section, we strive to understand the ideas generated by the following

More information

Section 6-3 Double-Angle and Half-Angle Identities

Section 6-3 Double-Angle and Half-Angle Identities 6-3 Double-Angle and Half-Angle Identities 47 Section 6-3 Double-Angle and Half-Angle Identities Double-Angle Identities Half-Angle Identities This section develops another important set of identities

More information

Physics 201 Homework 8

Physics 201 Homework 8 Physics 201 Homework 8 Feb 27, 2013 1. A ceiling fan is turned on and a net torque of 1.8 N-m is applied to the blades. 8.2 rad/s 2 The blades have a total moment of inertia of 0.22 kg-m 2. What is the

More information

1. [20 Points] Evaluate each of the following limits. Please justify your answers. Be clear if the limit equals a value, + or, or Does Not Exist.

1. [20 Points] Evaluate each of the following limits. Please justify your answers. Be clear if the limit equals a value, + or, or Does Not Exist. Answer Key, Math, Final Eamination, December 9, 9. [ Points] Evaluate each of the following limits. Please justify your answers. Be clear if the limit equals a value, + or, or Does Not Eist. (a lim + 6

More information

FOURIER TRANSFORM TERENCE TAO

FOURIER TRANSFORM TERENCE TAO FOURIER TRANSFORM TERENCE TAO Very broaly speaking, the Fourier transform is a systematic way to ecompose generic functions into a superposition of symmetric functions. These symmetric functions are usually

More information

15.2. First-Order Linear Differential Equations. First-Order Linear Differential Equations Bernoulli Equations Applications

15.2. First-Order Linear Differential Equations. First-Order Linear Differential Equations Bernoulli Equations Applications 00 CHAPTER 5 Differential Equations SECTION 5. First-Orer Linear Differential Equations First-Orer Linear Differential Equations Bernoulli Equations Applications First-Orer Linear Differential Equations

More information

TOPIC 4: DERIVATIVES

TOPIC 4: DERIVATIVES TOPIC 4: DERIVATIVES 1. The derivative of a function. Differentiation rules 1.1. The slope of a curve. The slope of a curve at a point P is a measure of the steepness of the curve. If Q is a point on the

More information

Some Notes on Taylor Polynomials and Taylor Series

Some Notes on Taylor Polynomials and Taylor Series Some Notes on Taylor Polynomials and Taylor Series Mark MacLean October 3, 27 UBC s courses MATH /8 and MATH introduce students to the ideas of Taylor polynomials and Taylor series in a fairly limited

More information

Lecture 7 : Inequalities 2.5

Lecture 7 : Inequalities 2.5 3 Lecture 7 : Inequalities.5 Sometimes a problem may require us to find all numbers which satisfy an inequality. An inequality is written like an equation, except the equals sign is replaced by one of

More information

PSS 27.2 The Electric Field of a Continuous Distribution of Charge

PSS 27.2 The Electric Field of a Continuous Distribution of Charge Chapter 27 Solutions PSS 27.2 The Electric Field of a Continuous Distribution of Charge Description: Knight Problem-Solving Strategy 27.2 The Electric Field of a Continuous Distribution of Charge is illustrated.

More information

Data Center Power System Reliability Beyond the 9 s: A Practical Approach

Data Center Power System Reliability Beyond the 9 s: A Practical Approach Data Center Power System Reliability Beyon the 9 s: A Practical Approach Bill Brown, P.E., Square D Critical Power Competency Center. Abstract Reliability has always been the focus of mission-critical

More information

f(x + h) f(x) h as representing the slope of a secant line. As h goes to 0, the slope of the secant line approaches the slope of the tangent line.

f(x + h) f(x) h as representing the slope of a secant line. As h goes to 0, the slope of the secant line approaches the slope of the tangent line. Derivative of f(z) Dr. E. Jacobs Te erivative of a function is efine as a limit: f (x) 0 f(x + ) f(x) We can visualize te expression f(x+) f(x) as representing te slope of a secant line. As goes to 0,

More information

Purpose of the Experiments. Principles and Error Analysis. ε 0 is the dielectric constant,ε 0. ε r. = 8.854 10 12 F/m is the permittivity of

Purpose of the Experiments. Principles and Error Analysis. ε 0 is the dielectric constant,ε 0. ε r. = 8.854 10 12 F/m is the permittivity of Experiments with Parallel Plate Capacitors to Evaluate the Capacitance Calculation an Gauss Law in Electricity, an to Measure the Dielectric Constants of a Few Soli an Liqui Samples Table of Contents Purpose

More information

Answer Key for the Review Packet for Exam #3

Answer Key for the Review Packet for Exam #3 Answer Key for the Review Packet for Eam # Professor Danielle Benedetto Math Ma-Min Problems. Show that of all rectangles with a given area, the one with the smallest perimeter is a square. Diagram: y

More information

Review of Fundamental Mathematics

Review of Fundamental Mathematics Review of Fundamental Mathematics As explained in the Preface and in Chapter 1 of your textbook, managerial economics applies microeconomic theory to business decision making. The decision-making tools

More information

Kater Pendulum. Introduction. It is well-known result that the period T of a simple pendulum is given by. T = 2π

Kater Pendulum. Introduction. It is well-known result that the period T of a simple pendulum is given by. T = 2π Kater Penulum ntrouction t is well-known result that the perio of a simple penulum is given by π L g where L is the length. n principle, then, a penulum coul be use to measure g, the acceleration of gravity.

More information

Slope and Rate of Change

Slope and Rate of Change Chapter 1 Slope and Rate of Change Chapter Summary and Goal This chapter will start with a discussion of slopes and the tangent line. This will rapidly lead to heuristic developments of limits and the

More information

Solutions to old Exam 1 problems

Solutions to old Exam 1 problems Solutions to old Exam 1 problems Hi students! I am putting this old version of my review for the first midterm review, place and time to be announced. Check for updates on the web site as to which sections

More information

Supporting Australian Mathematics Project. A guide for teachers Years 11 and 12. Calculus: Module 15. The calculus of trigonometric functions

Supporting Australian Mathematics Project. A guide for teachers Years 11 and 12. Calculus: Module 15. The calculus of trigonometric functions Supporting Australian Mathematics Project 3 4 5 6 7 8 9 0 A guie for teachers Years an Calculus: Moule 5 The calculus of trigonometric functions The calculus of trigonometric functions A guie for teachers

More information

REVIEW SHEETS INTERMEDIATE ALGEBRA MATH 95

REVIEW SHEETS INTERMEDIATE ALGEBRA MATH 95 REVIEW SHEETS INTERMEDIATE ALGEBRA MATH 95 A Summary of Concepts Needed to be Successful in Mathematics The following sheets list the key concepts which are taught in the specified math course. The sheets

More information

As customary, choice (a) is the correct answer in all the following problems.

As customary, choice (a) is the correct answer in all the following problems. PHY2049 Summer 2012 Instructor: Francisco Rojas Exam 1 As customary, choice (a) is the correct answer in all the following problems. Problem 1 A uniformly charge (thin) non-conucting ro is locate on the

More information

Trigonometry Review with the Unit Circle: All the trig. you ll ever need to know in Calculus

Trigonometry Review with the Unit Circle: All the trig. you ll ever need to know in Calculus Trigonometry Review with the Unit Circle: All the trig. you ll ever need to know in Calculus Objectives: This is your review of trigonometry: angles, six trig. functions, identities and formulas, graphs:

More information

Class 34. Interference of Light. Phase of a Wave. In Phase. 180 o Out of Phase. Constructive and Destructive Interferences.

Class 34. Interference of Light. Phase of a Wave. In Phase. 180 o Out of Phase. Constructive and Destructive Interferences. Phase of a Wave Interference of Light In interference phenomena, we often compare the phase ifference between two waves arriving at the same point from ifferent paths or sources. The phase of a wave at

More information

= f x 1 + h. 3. Geometrically, the average rate of change is the slope of the secant line connecting the pts (x 1 )).

= f x 1 + h. 3. Geometrically, the average rate of change is the slope of the secant line connecting the pts (x 1 )). Math 1205 Calculus/Sec. 3.3 The Derivative as a Rates of Change I. Review A. Average Rate of Change 1. The average rate of change of y=f(x) wrt x over the interval [x 1, x 2 ]is!y!x ( ) - f( x 1 ) = y

More information

9.1 Trigonometric Identities

9.1 Trigonometric Identities 9.1 Trigonometric Identities r y x θ x -y -θ r sin (θ) = y and sin (-θ) = -y r r so, sin (-θ) = - sin (θ) and cos (θ) = x and cos (-θ) = x r r so, cos (-θ) = cos (θ) And, Tan (-θ) = sin (-θ) = - sin (θ)

More information

Review Sheet for Third Midterm Mathematics 1300, Calculus 1

Review Sheet for Third Midterm Mathematics 1300, Calculus 1 Review Sheet for Third Midterm Mathematics 1300, Calculus 1 1. For f(x) = x 3 3x 2 on 1 x 3, find the critical points of f, the inflection points, the values of f at all these points and the endpoints,

More information

CHAPTER FIVE. Solutions for Section 5.1. Skill Refresher. Exercises

CHAPTER FIVE. Solutions for Section 5.1. Skill Refresher. Exercises CHAPTER FIVE 5.1 SOLUTIONS 265 Solutions for Section 5.1 Skill Refresher S1. Since 1,000,000 = 10 6, we have x = 6. S2. Since 0.01 = 10 2, we have t = 2. S3. Since e 3 = ( e 3) 1/2 = e 3/2, we have z =

More information

Review of Intermediate Algebra Content

Review of Intermediate Algebra Content Review of Intermediate Algebra Content Table of Contents Page Factoring GCF and Trinomials of the Form + b + c... Factoring Trinomials of the Form a + b + c... Factoring Perfect Square Trinomials... 6

More information

Trigonometric Identities and Conditional Equations C

Trigonometric Identities and Conditional Equations C Trigonometric Identities and Conditional Equations C TRIGONOMETRIC functions are widely used in solving real-world problems and in the development of mathematics. Whatever their use, it is often of value

More information

Blue Pelican Calculus First Semester

Blue Pelican Calculus First Semester Blue Pelican Calculus First Semester Teacher Version 1.01 Copyright 2011-2013 by Charles E. Cook; Refugio, Tx Edited by Jacob Cobb (All rights reserved) Calculus AP Syllabus (First Semester) Unit 1: Function

More information