(x) = lim. x 0 x. (2.1)

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "(x) = lim. x 0 x. (2.1)"

Transcription

1 Differentiation. Derivative of function Let us fi an arbitrarily chosen point in the domain of the function y = f(). Increasing this fied value by we obtain the value of independent variable +. The value of function at this point is f(+). The change of the function y = f(+) f() is called the increment of the function which corresponds to the increment of the independent variable. The average rate of change of y with respect to from to + is y = f( + ) f() Definition. The limit of the average rate of change of y as approaches zero is called the derivative of f() at and denoted f (). According to this definition f y () 0. (.) The derivative of the function is denoted f () or y. These are Newton s notations of derivative. Also there is widely used Leibnitz s notation dy d or df d. Definition The function which has derivative at is called differentiable at. In other words, the function is differentiable at if there eists the limit (.). Consider the geometrical and physical concept to the derivative. For geometrical concept we use the graph of the function (Figure.). To the fied value there corresponds the point P on the graph of the function and to the variable value + the point Q. Let us denote the angle of elevation of the secant line through points P and Q by φ. In the right triangle P RQ the angle at P is also φ (these are conjugate angles). The length of the opposite side RQ of this angle is y and the length of the adjacent side P R is. Thus, the ratio of the increments of dependent variable and independent variable tan φ = y equals to the slope of the secant line P Q. As 0, i.e. +, the point P remains fied, Q moves along the curve toward P, and the secant

2 y f( + ) Q f() P R α φ + Figure.: Geometrical concept of derivative line through PQ changes its direction in such a way that its slope approaches the slope of the tangent line drawn to the graph at P. When α denotes the angle of elevation of tangent line at P then f y () 0 tan φ = tan α φ α Consequently, geometrically the derivative is the slope of the tangent line drawn to the graph of the function at the point with abscissa. If we treat the variable as time variable, then the function y = f() describes some mechanical or physical process, for eample movement along straight line. At the fied time the moving particle is in the position f() and at the time + in the position f( + ). During time interval the particle has passed the distance y. In this case the ratio y is an average velocity of this particle during the time interval. If gets smaller, the average velocity gets closer to the velocity the particle has at a fied time. Thus the limit, as approaches 0, i.e. the derivative of the movement is an instantaneous velocity at a time. When f() describes the position of the moving particle then f () describes the instantaneous velocity of this moving particle. More generally, if f() describes some mechanical, physical etc. process then the derivative f () describes the instantaneous rate of change of this process.

3 . Continuity and differentiability The purpose of this subsection is to show that from differentiability of a function at a point it always follows continuity of the function at this point. The converse assertion is not true. Theorem.. If the function y = f() is differentiable at, then it is continuous at. Proof. Let the function y = f() be differentiable at, that means f y (). To obtain the necessary and sufficient condition of the 0 continuity of the function holds, we find y y lim y lim = f () 0 = 0, 0 what is we wanted to prove. The following eample proves that the converse assertion does not hold. The function y = is continuous at = 0 but not differentiable at this point. The increment of the function at = 0 is Hence y = = lim y = i.e. there holds the necessary and sufficient condition of continuity at = 0. From evaluation of one-sided limits at = 0 and lim 0 = lim 0+ = it follows that there does not eist the limit lim 0 function y = does not have derivative at = 0., that means the.3 Derivatives of some basic elementary functions In this subsection we find the derivatives of basic elementary functions using the definition of the derivative (.). Let us start with the constant function y = c. In this case f() = c and for 0 f( + ) = c thus 3

4 y = c c = 0. The derivative of a constant c 0 = 0. Here we 0 have the first rule: the derivative of any constant equals to zero: c = 0 Second we find the derivative of a power function with an arbitrary natural eponent y = n. In this case f() = n, f( + ) = ( + ) n and the increment of the function y = ( + ) n n. By Newton s binomial formula y = n + n n + C n n n n = Dividing this equality by we have n n + C n n n. y = nn + C n n n and the limit ( n ) y 0 = nn because all the terms starting from second contain some power of with positive eponent. So, the derivative of the power function with natural eponent ( n ) = n n (.) Third we find the derivative of square root function y =. We find the increment of the function y = + and according to the definition of the derivative ( ) 0 lim ( + + ) 0 ( + )( + + ) 0 ( + + ) = + + = = Fourth we find the derivative of reciprocal function y =. The increment of the function is y = + = ( + ) = ( + ) and due to the definition of the derivative ( ) 0 ( + ) = = 4.

5 The last two derivatives give an idea that the formula of the derivative of power function (.) holds not only for natural eponents but also for negative integer and fractional eponents. This fact we can prove later. Fifth we find the derivative of sine function y = sin. We find the increment of the function y = sin(+) sin = sin cos +cos sin sin = cos sin sin ( cos ). By the definition of the derivative of the function (here we use some properties of the limits) (sin ) cos sin sin ( cos ) = 0 ( ) cos sin sin ( cos ) = 0 sin cos = cos lim sin lim = 0 0 ( cos )( + cos ) = cos sin lim = 0 ( + cos ) Thus, = cos sin lim 0 = cos sin lim 0 sin ( + cos ) = sin sin + cos (sin ) = cos As a result of similar transformations one can prove that (cos ) = sin = cos sin 0 = cos. Seventh we find the derivative of the natural logarithm y = ln. We fi a value of in the domain of this function > 0 and find the increment of the function y = ln( + ) ln = ln + ( = ln + ). By the definition of the derivative ( (ln ) 0 ln + ) ( ln + ). 0 The value > 0 is fied and as 0, ± and (ln ) ( ln ln + [ ( + 5 ) = ) ] = ln e =.

6 Consequently (ln ) = The derivatives of the rest basic elementary function we find in the following subsections..4 Rules of differentiation The finding a derivative of a function is called differentiation of given function. So, the rules of the differentiation are the rules of finding derivatives. Let us have two functions u = u() and v = v(). Assume that both of them are differentiable at. Theorem 4. (the sum rule). The derivative of the sum of the functions equals to the sum of derivatives of these functions: [u() + v()] = u () + v () Proof. Denote the sum y() = u() + v(). The increment of this sum y = u( + ) + v( + ) [u() + v()] = = u( + ) u() + v( + ) v() = u + v and with help the corresponding property of limits y u + v () 0 u 0 + lim v 0 = u () + v (). Eample 4.. Find the derivative of the function y = + sin By the sum rule y = ( + sin ) = + sin ) = + cos. This rule also applies to additions of more than two functions [u() + v() + w() +...] = u () + v () + w () +... Eample 4.. Find the derivative of the function y = By the sum rule y = ( ) = ( 3 ) +( ) + + = = 3 ++ Theorem 4. (the product rule). The derivative of the product of two functions [u()v()] = u ()v() + u()v () 6

7 Proof. Let us denote the product y() = u()v(). Then y = u( + )v( + ) u()v() = = u( + )v( + ) u()v( + ) + u()v( + ) u()v() = = [u( + ) u()]v( + ) + u()[v( + ) v()] = = u v( + ) + u() v. By the properties of the limits y u v( + ) + u() v () 0 0 = u lim v( + ) + u() lim 0 0 v. As we assumed, the function v() is differentiable at. The theorem. states that v() is also continuous at. According to the third condition of continuity lim v( + ) = v(). It follows that 0 y () = u ()v() + u()v (). Eample 4.3. Find the derivative of the function y = sin + cos y = ( sin + cos ) = ( sin ) + (cos ) = = sin + (sin ) sin = sin + cos sin = cos. Conclusion 4.3 (the constant factor rule). The constant factor can be taken outside the derivative: [c u()] = c u () Indeed, by theorem 4. [c u()] = c u()+c u () = 0 u()+c u () = c u (). Using this constant factor rule we find the derivative of the net basic elementary function y = log a (a > 0, a ). Here we use the change of base formula for logarithms log a = ln. We obtain (log a ) = ln a ( ) ln a ln = ln a (ln ) = ln a = ln a So we have a new formula of differentiation (log a ) = ln a 7

8 Conclusion 4.4 (the subtraction rule). The derivative of the difference of two functions equals to the difference of the derivatives of these functions: [u() v()] = u () v (). To verify it we use the theorem 4. and conclusion 4.3: [u() v()] = [u() + ( )v()] = u () + [( )v()] = u () v (). Theorem 4.5 (the quotient rule). The derivative of the quotient [ ] u() = u ()v() u()v () v() v () provided v() 0. Proof. Denote the quotient y() = u() v() and, as v() 0, then u() = y()v(). By the product rule hence Substituting y() we obtain y () = u () = y ()v() + y()v () y () = u () y()v () v() u () u()v () v() v() = u ()v() u()v () v () which is we wanted to prove. Using the theorem 4.5 we find the derivative of y = tan (tan ) = ( ) sin = (sin ) cos sin (cos ) cos cos = cos + sin cos = cos Thus, (tan ) = cos As well one can prove with help the theorem 4.5 that (cot ) = sin 8

9 .5 Derivative of inverse function Here we restrict ourselves to one-to-one function y = f(), which has inverse function = f (y). Theorem 5.. If the function y = f() has a nonzero derivative at f () 0, then the derivative of inverse function (f (y)) = f () Proof. The independent variable of the inverse function is y. By definition the derivative is (f (y)) y 0 y = y lim y 0 According to the assumption y = f() is differentiable at, hence by the theorem. continuous at. The inverse function = f (y) of a continuous function is also continuous at y. Thus, y 0 yields 0. Therefore (f (y)) = lim 0 y = f () what is we wanted to prove. We can read the assertion of the theorem 5. as follows: the derivative of the inverse function equals to reciprocal of the derivative of given function. The opposite also holds. The derivative of the given function is the reciprocal of the derivative of the inverse function: f () = (f (y)). (.3) The last equality we shall use to find some more derivatives of basic elementary functions. Let us start from eponential function y = a, (a > 0, a ). The inverse function of this function is = log a y and by (.3) (a ) = (log a y) = y ln a The derivative of the eponential function (a ) = a ln a = y ln a = a ln a 9

10 In case of eponential function y = e there holds ln e = and we obtain a special case of the last formula (e ) = e Net we find the derivative of y = arcsin. The inverse function is = [ sin y. The range of y = arcsin and also the domain of inverse function is π ; π ]. Therefore cos y 0 and by (.3) (arcsin ) = = (sin y) cos y = sin y = Thus, (arcsin ) = It is known that [ ; ] arcsin + arccos = π,. It follows, that arccos = π arcsin and as π is a constant, we have (arccos ) = Net we find the derivative of y = arctan. = tan y and by (.3) The inverse function is (arctan ) = (tan y) = cos y = + tan y = +. Consequently (arctan ) = + For each real arctan + arccot = π, hence (arccot ) = + 0

11 .6 Derivative of the composite function The components of the composite function y = f[φ()] are y = f(u) and u = φ(). Theorem 6.. If u = φ() is differentiable at and y = f(u) is differentiable at u, then the composite function y = f[φ()] is differentiable at and {f[φ()]} = f [φ()]φ (). (.4) Proof. Denote the composite function F () = f[φ()]. Then y = F () and the derivative F y () y u lim 0 y u u = u. The function u = φ() is differentiable at. By theorem. it is also continuous at. It follows that there holds the necessary and sufficient condition of continuity, i.e. 0 yields u 0. Therefore F y () u 0 u lim u 0 = f (u)φ () which is we wanted to prove. The equality (.4) is also referred as chain rule. With help of this rule we find now the derivative of the common power function y = α, where α is whatever real eponent and > 0. We epress the power function as and by (.4) α = e α ln ( α ) = ( e α ln ) = e α ln (α ln ) = e α ln α = α α = αα Thus, for every real α whenever α is defined ( α ) = α α Using ( e ) = e ( ) = e, we have (sinh ) = ( ) (e e ) = (e + e ) = cosh

12 or In similar way one can prove (sinh ) = cosh (cosh ) = sinh According to the quotient rule we find ( ) sinh (tanh ) = = (sinh ) cosh sinh (cosh ) cosh cosh Thus, As well one can prove that (tanh ) = cosh (coth ) = sinh = cosh sinh cosh = cosh.7 Implicit differentiation A method called implicit differentiation makes use of the chain rule to differentiate implicitly defined functions F (, y) = 0. One possibility for differentiation of implicit function is to write it in eplicit form (if one can solve the equation for y) y = f() and then differentiate it using the rules considered in previous subsections. Often the implicit functions are multi-valued and then we need to differentiate every unique branch. In many cases the transformation of the function given implicitly to an eplicit form can be complicated. Sometimes it is impossible to epress y eplicitly as a function of and therefore y cannot be found by eplicit differentiation. Consider some eamples of implicit differentiation. Eample 7.. Find y, if + y = r. We differentiate both sides of this equality with respect to, taking into account that y is a composite function: y is a function of and the square function is the function of y. On the right side of the equality is the constant r. Using chain rule for y we have the result of the differentiation Solving this equation for y we obtain + y y = 0 y = y

13 In order to differentiate function this eplicitly, one have to solve the equation with respect to y y = ± r. The derivative of the first unique branch y = r is y = r ( ) = r The derivative of the second unique branch y = r is y = r ( ) = r In both cases the result matches with the result of the implicit differentiation. Eample 7.. Find y, if sin( + y) + cos(y) = 0. On the left side of this equality we have two composite functions. In the first the eternal function is sine function and internal function is + y, in the second eternal function is cosine function and internal function y. If we differentiate both sides of the equality with respect to, we have Removing parenthesis gives cos( + y) ( + y ) sin(y) (y + y ) = 0 cos( + y) + y cos( + y) y sin(y) y sin(y) = 0 or and y [cos( + y) sin(y)] = y sin(y) cos( + y) y = y sin(y) cos( + y) cos( + y) sin(y).8 Logarithmic differentiation Logarithmic differentiation is a means of differentiating algebraically complicated functions or functions for which the ordinary rules of differentiation do not apply. First of all we have to use logarithmic differentiation if we have the function y = [f()] g(), i.e. if we have variable base to variable eponent. In power function the eponent has to be constant and in eponential function the base has to be constant. So, we cannot treat this kind of function as power function (because the eponent g() depends on ). Also we cannot treat this function as eponential function (because f() depends on ). 3

14 Taking a logarithm of such function gives us possibility to transform the function so that the ordinary rules of differentiation will be applicable. If we take a logarithm to base e, we have ln y = ln[f()] g() = g() ln f() and [f()] g() has been transformed to product and we have the possibility to differentiate it using the product rule and the chain rule. The variable y is the function of the independent variable, so the left side of the equality ln y is composite function and its derivative (ln y) = y y Eample 8.. Find the derivative of y = ( + ). Taking a natural logarithm gives us ln y = ln( + ) and the differentiation of this equality y y = ln( + ) + + or y y = ln( + ) + + Multiplying both sides of the last equality by y, we have ) y = y (ln( + ) + + and after substituting y we obtain the desired derivative ( ) y = ( + ) ln( + ) + + Eample 8.. Find the derivative of the function y = 3. ( + 3) The derivative of this function can be found without logarithmic differentiation. It is possible to differentiate this function by applying the quotient rule, the product rule and the chain rule. This would be quite complicated. We can simplify the differentiation by taking logarithms of both sides. We need to use the properties of logarithms to epand the right side as follows ln y = ln 3 = 3 ln + 5 ( + 3) ln( ) ln( + 3) 5 4 5

15 Now the differentiation process will be simpler: y y = Multiplying both sides of equality by y [ ] 3 y = y + ( ) 5( + 3) and substituting y we have the result y = 3 [ ] 3 5 ( + 3) + ( ) 5( + 3).9 Parametric differentiation Let us consider the parametric representation of a function { = φ(t) y = ψ(t). Assume that both functions of the variable t are one-valued and differentiable and the derivative of with respect to t d 0 and = φ(t) has one-valued dt inverse function t = Φ(). The variable y is composite function with respect to, i.e. y = ψ[φ()] and by chain rule dy d = ψ [Φ()] Φ () (.5) According to the rule of the derivative of the inverse function Φ () = φ (t) = d dt Using notations ψ [Φ()] = ψ (t) = dy, we obtain from (.5) dt dy dy d = dt d dt 5

16 In calculus the derivative by parameter is denoted d dt = ẋ and read -dot. As well dy dt = ẏ which is read y-dot. We can conclude that the derivative of a function given in parametric form is dy d = ẏ (.6) ẋ Eample 9.. The parametric representation of the implicit function + y = r is { = r cos t y = r sin t. If we find the derivatives of both functions with respect to parameter t, we have ẋ = r sin t and ẏ = r cos t and by the rule (.6) we obtain the result dy d cos t = r r sin t = cos t sin t or dy d = y which is the same we have got in Eample 7.. Eample 9.. Find the slope of the tangent line of cycloid { = a(t sin t) y = a( cos t) at the point that corresponds to the value of parameter t = π. The slope of the tangent line of the curve (the graph of the function) at a point equals to the value of derivative at this point. Therefore we need to evaluate the derivative at the point with t = π. To obtain that we find ẋ = a( cos t) and ẏ = a sin t, then by (.6) dy d = a sin t a( cos t) = sin t cos t The value of the derivative at the point where t = π, is sin π cos π = Consequently the slope of the tangent line at the point, where t = π, equals to. 6

17 .0 Differential of function In many cases it is sufficient to consider instead of the increment of a function its linear part (sometimes called principal part). The derivative of the function y = f() at the point is defined as But then the variable y f y () 0 is the sum y = f () + α where α is an infinitesimal as 0. Multiplying both sides of the last equality by, we have y = f () + α (.7) On the right side of equality (.7) the first addend is for fied value of linear with respect to, but the second addend is an infinitesimal of a higher order with respect to because α lim 0 = 0 Definition. The linear part f () of the increment of the function (.7) is called the differential of the function and denoted dy. Due to this definition dy = f () If the independent and dependent variable are equal, i,e, y = then y = and dy = d =. Consequently for independent variable there holds d =, that means for the independent variable two concepts increment and differential coincide. Hence, the differential of the function f() can be written dy = f ()d (.8) Eample 0.. Find the epression of the differential of the function y = arctan. By the chain rule y = + and using (.8) we write the epression of the differential dy = + d = d ( + ) 7

18 Eample 0.. Evaluate the increment and the differential of the function y = if the independent variable increases from to, 05. First we find the increment of the function y =, 05 = 0, 05 The increment of the independent variable is d = = 0, 05 the derivative y = thus, the value of the differential dy = 0, 05 = 0,. Net let us find out, what is the geometric interpretation of the differential of the function. y f( + ) Q f() P T R } y } dy + Figure.: the differential of the function The derivative of the function is the slope of the tangent line drawn to the graph at the point P (with abscissa ) or the tangent of the angle of elevation. The product f ()d equals to the length of the side RT of the right triangle P RT, i.e. the differential of the function is the length of the interval RT. Consequently the differential of the function means the increment of y as increases by, if the movement along the curve has been replaced with the movement along the tangent line. Mechanically the derivative means the instantaneous velocity and this velocity is a variable. If we fi a time, we fi f (), i.e. the instantaneous velocity of the moving object. If this moving object keeps on moving with this constant speed, the product f ()d equals to the distance this moving object has passed during the time interval with that constant speed. If is sufficiently small then, since the difference of y and dy is a quantity, which is the infinitesimal of a higher order with respect to, 8

19 we may write y dy. Using the definitions of the increment and the differential of the function, we get which yields the approimate formula f( + ) f() f () f( + ) f() + f (). (.9) The formula (.9) is applicable only for relatively small increments of the argument. Eample 0.3. Using the formula (.9) let us calculate the approimate value of ln 0, 9. Here we choose =, = 0, and the function f() = ln. The value of the function at = equals to f() = ln = 0 and the value of the derivative f () = at = equals to f () =. Therefore, by (.9) we get the approimate value ln 0, ( 0, ) = 0, which differs from the eact value less than 0, Higher order derivatives The derivative f () of the function y = f() is the function of the independent variable again and it is possible to differentiate this derivative. Definition.. The derivative of the derivative of the function y = f() is called second derivative and denoted f (), that is f () = [f ()] The second order derivative is denoted also y. The Leibnitz notation is d y d = d ( ) dy d d which we read d squared y d squared or d f d Eample.. Let us find the second derivative of the function y = e. First using the chain rule we find the derivative y = e ( ) = e and net using the product rule and the chain rule we find the second derivative y = e e ( ) = e ( ) 9

20 The third derivative of the function y = f() is defined as the derivative of the second derivative, i.e. or by Leibnitz notation f () = [f ()] d 3 y d = d 3 d ( ) d y Definition.. The nth order derivative f (n) () of the function y = f() is defined as the derivative of the n -st order derivative: By Leibnitz notation d f (n) () = [ f (n ) () ] d d ( ) d n y d n Eample.. Find the nth order derivative of the sine function y = sin. To find the nth order derivative we use the mathematical induction. First we form the base of the induction finding the few first derivatives y = ( cos = sin + π ), ( y = sin = sin( + π) = sin y = ( cos = sin + 3 π ), ( y (4) = sin = sin( + π) = sin + π + 4 π ( Now we are able to form the induction hypothesis y (n) = sin prove it we find the n + -st order derivative ), ). + n π ). To. ( y (n+) = cos + n π ) ( = sin + nπ + π ) ( = sin + (n + ) π ). Equations of tangent and normal lines of curve As in previous subsections the curve is a graph of a function. 0

21 We shall use the point-slope equation of the line. If the line passes the point P 0 ( 0 ; y 0 ) and has the slope k, then the equation of this line is y y 0 = k( 0 ) The ordinate of the point with abscissa 0 on the graph of the function y = f() is f( 0 ). The slope of the tangent line is f ( 0 ). Thus, according the point-slope equation we get the equation of the tangent line y f( 0 ) = f ( 0 )( 0 ). (.0) Definition.. The perpendicular of the tangent line of the curve at a point is called the normal line or normal of the curve at this point. y normal y = f() tangent f( 0 ) 0 Figure.3: the tangent and normal line of the curve If two lines are perpendicular and the slopes of the first and second lines are k and k, respectively, then these slopes satisfy the condition k k =. Hence, if we know the slope k of the given line, then the slope of the perpendicular line is k = k. Therefore, the slope of the normal line at the point with abscissa 0 is and the normal has the equation f ( 0 ) y f( 0 ) = f ( 0 ) ( 0). (.)

22 Eample. Find the equations of tangent and normal lines of the function y = cos at the point with abscissa 0 = π 6. In our case the ordinate of the given point is f( 0 ) = cos π 3 6 =. Using the derivative f () = sin, we find the the slope of the tangent line ( π ) f = and the slope of the normal line 6 f ( π) =. 6 The tangent line has the equation 3 ( y = π ) 6 or y = + π The approimate value of the intercept of this line is π + 6 3, 8. The normal has the equation 3 y ( = π ) 6 or y = π 6 The approimate value of the intercept of this line is 3 3 π 6 0, 8.

23 tangent y normal 3 π π 6 π y = cos Figure(.4: the tangent and the normal lines of the function y = cos at the ) π 3 point 6 ; 3

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

Algebra. Exponents. Absolute Value. Simplify each of the following as much as possible. 2x y x + y y. xxx 3. x x x xx x. 1. Evaluate 5 and 123

Algebra. Exponents. Absolute Value. Simplify each of the following as much as possible. 2x y x + y y. xxx 3. x x x xx x. 1. Evaluate 5 and 123 Algebra Eponents Simplify each of the following as much as possible. 1 4 9 4 y + y y. 1 5. 1 5 4. y + y 4 5 6 5. + 1 4 9 10 1 7 9 0 Absolute Value Evaluate 5 and 1. Eliminate the absolute value bars from

More information

Core Maths C3. Revision Notes

Core Maths C3. Revision Notes Core Maths C Revision Notes October 0 Core Maths C Algebraic fractions... Cancelling common factors... Multipling and dividing fractions... Adding and subtracting fractions... Equations... 4 Functions...

More information

UNIT 1: ANALYTICAL METHODS FOR ENGINEERS

UNIT 1: ANALYTICAL METHODS FOR ENGINEERS UNIT : ANALYTICAL METHODS FOR ENGINEERS Unit code: A/60/40 QCF Level: 4 Credit value: 5 OUTCOME 3 - CALCULUS TUTORIAL DIFFERENTIATION 3 Be able to analyse and model engineering situations and solve problems

More information

South Carolina College- and Career-Ready (SCCCR) Pre-Calculus

South Carolina College- and Career-Ready (SCCCR) Pre-Calculus South Carolina College- and Career-Ready (SCCCR) Pre-Calculus Key Concepts Arithmetic with Polynomials and Rational Expressions PC.AAPR.2 PC.AAPR.3 PC.AAPR.4 PC.AAPR.5 PC.AAPR.6 PC.AAPR.7 Standards Know

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

MATHEMATICS (CLASSES XI XII)

MATHEMATICS (CLASSES XI XII) MATHEMATICS (CLASSES XI XII) General Guidelines (i) All concepts/identities must be illustrated by situational examples. (ii) The language of word problems must be clear, simple and unambiguous. (iii)

More information

Items related to expected use of graphing technology appear in bold italics.

Items related to expected use of graphing technology appear in bold italics. - 1 - Items related to expected use of graphing technology appear in bold italics. Investigating the Graphs of Polynomial Functions determine, through investigation, using graphing calculators or graphing

More information

1 Review of complex numbers

1 Review of complex numbers 1 Review of complex numbers 1.1 Complex numbers: algebra The set C of complex numbers is formed by adding a square root i of 1 to the set of real numbers: i = 1. Every complex number can be written uniquely

More information

MEMORANDUM. All students taking the CLC Math Placement Exam PLACEMENT INTO CALCULUS AND ANALYTIC GEOMETRY I, MTH 145:

MEMORANDUM. All students taking the CLC Math Placement Exam PLACEMENT INTO CALCULUS AND ANALYTIC GEOMETRY I, MTH 145: MEMORANDUM To: All students taking the CLC Math Placement Eam From: CLC Mathematics Department Subject: What to epect on the Placement Eam Date: April 0 Placement into MTH 45 Solutions This memo is an

More information

Core Maths C2. Revision Notes

Core Maths C2. Revision Notes Core Maths C Revision Notes November 0 Core Maths C Algebra... Polnomials: +,,,.... Factorising... Long division... Remainder theorem... Factor theorem... 4 Choosing a suitable factor... 5 Cubic equations...

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

Mathematics 31 Pre-calculus and Limits

Mathematics 31 Pre-calculus and Limits Mathematics 31 Pre-calculus and Limits Overview After completing this section, students will be epected to have acquired reliability and fluency in the algebraic skills of factoring, operations with radicals

More information

Geometry Notes RIGHT TRIANGLE TRIGONOMETRY

Geometry Notes RIGHT TRIANGLE TRIGONOMETRY Right Triangle Trigonometry Page 1 of 15 RIGHT TRIANGLE TRIGONOMETRY Objectives: After completing this section, you should be able to do the following: Calculate the lengths of sides and angles of a right

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

Inverse Circular Function and Trigonometric Equation

Inverse Circular Function and Trigonometric Equation Inverse Circular Function and Trigonometric Equation 1 2 Caution The 1 in f 1 is not an exponent. 3 Inverse Sine Function 4 Inverse Cosine Function 5 Inverse Tangent Function 6 Domain and Range of Inverse

More information

Algebra and Geometry Review (61 topics, no due date)

Algebra and Geometry Review (61 topics, no due date) Course Name: Math 112 Credit Exam LA Tech University Course Code: ALEKS Course: Trigonometry Instructor: Course Dates: Course Content: 159 topics Algebra and Geometry Review (61 topics, no due date) Properties

More information

2008 AP Calculus AB Multiple Choice Exam

2008 AP Calculus AB Multiple Choice Exam 008 AP Multiple Choice Eam Name 008 AP Calculus AB Multiple Choice Eam Section No Calculator Active AP Calculus 008 Multiple Choice 008 AP Calculus AB Multiple Choice Eam Section Calculator Active AP Calculus

More information

Chapter 7 Outline Math 236 Spring 2001

Chapter 7 Outline Math 236 Spring 2001 Chapter 7 Outline Math 236 Spring 2001 Note 1: Be sure to read the Disclaimer on Chapter Outlines! I cannot be responsible for misfortunes that may happen to you if you do not. Note 2: Section 7.9 will

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

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

Substitute 4 for x in the function, Simplify.

Substitute 4 for x in the function, Simplify. Page 1 of 19 Review of Eponential and Logarithmic Functions An eponential function is a function in the form of f ( ) = for a fied ase, where > 0 and 1. is called the ase of the eponential function. The

More information

Introduction to Calculus

Introduction to Calculus Introduction to Calculus Contents 1 Introduction to Calculus 3 11 Introduction 3 111 Origin of Calculus 3 112 The Two Branches of Calculus 4 12 Secant and Tangent Lines 5 13 Limits 10 14 The Derivative

More information

Precalculus REVERSE CORRELATION. Content Expectations for. Precalculus. Michigan CONTENT EXPECTATIONS FOR PRECALCULUS CHAPTER/LESSON TITLES

Precalculus REVERSE CORRELATION. Content Expectations for. Precalculus. Michigan CONTENT EXPECTATIONS FOR PRECALCULUS CHAPTER/LESSON TITLES Content Expectations for Precalculus Michigan Precalculus 2011 REVERSE CORRELATION CHAPTER/LESSON TITLES Chapter 0 Preparing for Precalculus 0-1 Sets There are no state-mandated Precalculus 0-2 Operations

More information

Indiana State Core Curriculum Standards updated 2009 Algebra I

Indiana State Core Curriculum Standards updated 2009 Algebra I Indiana State Core Curriculum Standards updated 2009 Algebra I Strand Description Boardworks High School Algebra presentations Operations With Real Numbers Linear Equations and A1.1 Students simplify and

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

Core Maths C1. Revision Notes

Core Maths C1. Revision Notes Core Maths C Revision Notes November 0 Core Maths C Algebra... Indices... Rules of indices... Surds... 4 Simplifying surds... 4 Rationalising the denominator... 4 Quadratic functions... 4 Completing the

More information

Roots and Coefficients of a Quadratic Equation Summary

Roots and Coefficients of a Quadratic Equation Summary Roots and Coefficients of a Quadratic Equation Summary For a quadratic equation with roots α and β: Sum of roots = α + β = and Product of roots = αβ = Symmetrical functions of α and β include: x = and

More information

Dr. Foote s Algebra & Calculus Tips or How to make life easier for yourself, make fewer errors, and get more points

Dr. Foote s Algebra & Calculus Tips or How to make life easier for yourself, make fewer errors, and get more points Dr. Foote s Algebra & Calculus Tips or How to make life easier for yourself, make fewer errors, and get more points This handout includes several tips that will help you streamline your computations. Most

More information

Section 5-9 Inverse Trigonometric Functions

Section 5-9 Inverse Trigonometric Functions 46 5 TRIGONOMETRIC FUNCTIONS Section 5-9 Inverse Trigonometric Functions Inverse Sine Function Inverse Cosine Function Inverse Tangent Function Summar Inverse Cotangent, Secant, and Cosecant Functions

More information

Chapter 11 - Curve Sketching. Lecture 17. MATH10070 - Introduction to Calculus. maths.ucd.ie/modules/math10070. Kevin Hutchinson.

Chapter 11 - Curve Sketching. Lecture 17. MATH10070 - Introduction to Calculus. maths.ucd.ie/modules/math10070. Kevin Hutchinson. Lecture 17 MATH10070 - Introduction to Calculus maths.ucd.ie/modules/math10070 Kevin Hutchinson 28th October 2010 Z Chain Rule (I): If y = f (u) and u = g(x) dy dx = dy du du dx Z Chain rule (II): d dx

More information

Estimated Pre Calculus Pacing Timeline

Estimated Pre Calculus Pacing Timeline Estimated Pre Calculus Pacing Timeline 2010-2011 School Year The timeframes listed on this calendar are estimates based on a fifty-minute class period. You may need to adjust some of them from time to

More information

Calculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum

Calculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum Calculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum UNIT I: The Hyperbolic Functions basic calculus concepts, including techniques for curve sketching, exponential and logarithmic

More information

Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks

Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Welcome to Thinkwell s Homeschool Precalculus! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson

More information

Exponential and Logarithmic Functions

Exponential and Logarithmic Functions Chapter 6 Eponential and Logarithmic Functions Section summaries Section 6.1 Composite Functions Some functions are constructed in several steps, where each of the individual steps is a function. For eample,

More information

MyMathLab ecourse for Developmental Mathematics

MyMathLab ecourse for Developmental Mathematics MyMathLab ecourse for Developmental Mathematics, North Shore Community College, University of New Orleans, Orange Coast College, Normandale Community College Table of Contents Module 1: Whole Numbers and

More information

ALGEBRA I / ALGEBRA I SUPPORT

ALGEBRA I / ALGEBRA I SUPPORT Suggested Sequence: CONCEPT MAP ALGEBRA I / ALGEBRA I SUPPORT August 2011 1. Foundations for Algebra 2. Solving Equations 3. Solving Inequalities 4. An Introduction to Functions 5. Linear Functions 6.

More information

Graphing Trigonometric Skills

Graphing Trigonometric Skills Name Period Date Show all work neatly on separate paper. (You may use both sides of your paper.) Problems should be labeled clearly. If I can t find a problem, I ll assume it s not there, so USE THE TEMPLATE

More information

G. GRAPHING FUNCTIONS

G. GRAPHING FUNCTIONS G. GRAPHING FUNCTIONS To get a quick insight int o how the graph of a function looks, it is very helpful to know how certain simple operations on the graph are related to the way the function epression

More information

Georgia Department of Education Kathy Cox, State Superintendent of Schools 7/19/2005 All Rights Reserved 1

Georgia Department of Education Kathy Cox, State Superintendent of Schools 7/19/2005 All Rights Reserved 1 Accelerated Mathematics 3 This is a course in precalculus and statistics, designed to prepare students to take AB or BC Advanced Placement Calculus. It includes rational, circular trigonometric, and inverse

More information

6.2 Properties of Logarithms

6.2 Properties of Logarithms 6. Properties of Logarithms 437 6. Properties of Logarithms In Section 6., we introduced the logarithmic functions as inverses of eponential functions and discussed a few of their functional properties

More information

Microeconomic Theory: Basic Math Concepts

Microeconomic Theory: Basic Math Concepts Microeconomic Theory: Basic Math Concepts Matt Van Essen University of Alabama Van Essen (U of A) Basic Math Concepts 1 / 66 Basic Math Concepts In this lecture we will review some basic mathematical concepts

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

MATH 095, College Prep Mathematics: Unit Coverage Pre-algebra topics (arithmetic skills) offered through BSE (Basic Skills Education)

MATH 095, College Prep Mathematics: Unit Coverage Pre-algebra topics (arithmetic skills) offered through BSE (Basic Skills Education) MATH 095, College Prep Mathematics: Unit Coverage Pre-algebra topics (arithmetic skills) offered through BSE (Basic Skills Education) Accurately add, subtract, multiply, and divide whole numbers, integers,

More information

L 2 : x = s + 1, y = s, z = 4s + 4. 3. Suppose that C has coordinates (x, y, z). Then from the vector equality AC = BD, one has

L 2 : x = s + 1, y = s, z = 4s + 4. 3. Suppose that C has coordinates (x, y, z). Then from the vector equality AC = BD, one has The line L through the points A and B is parallel to the vector AB = 3, 2, and has parametric equations x = 3t + 2, y = 2t +, z = t Therefore, the intersection point of the line with the plane should satisfy:

More information

MPE Review Section III: Logarithmic & Exponential Functions

MPE Review Section III: Logarithmic & Exponential Functions MPE Review Section III: Logarithmic & Eponential Functions FUNCTIONS AND GRAPHS To specify a function y f (, one must give a collection of numbers D, called the domain of the function, and a procedure

More information

Higher Education Math Placement

Higher Education Math Placement Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication

More information

Section 1.1 Linear Equations: Slope and Equations of Lines

Section 1.1 Linear Equations: Slope and Equations of Lines Section. Linear Equations: Slope and Equations of Lines Slope The measure of the steepness of a line is called the slope of the line. It is the amount of change in y, the rise, divided by the amount of

More information

Calculus. Contents. Paul Sutcliffe. Office: CM212a.

Calculus. Contents. Paul Sutcliffe. Office: CM212a. Calculus Paul Sutcliffe Office: CM212a. www.maths.dur.ac.uk/~dma0pms/calc/calc.html Books One and several variables calculus, Salas, Hille & Etgen. Calculus, Spivak. Mathematical methods in the physical

More information

2 Complex Functions and the Cauchy-Riemann Equations

2 Complex Functions and the Cauchy-Riemann Equations 2 Complex Functions and the Cauchy-Riemann Equations 2.1 Complex functions In one-variable calculus, we study functions f(x) of a real variable x. Likewise, in complex analysis, we study functions f(z)

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

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

Roots of Equations (Chapters 5 and 6)

Roots of Equations (Chapters 5 and 6) Roots of Equations (Chapters 5 and 6) Problem: given f() = 0, find. In general, f() can be any function. For some forms of f(), analytical solutions are available. However, for other functions, we have

More information

Prentice Hall Mathematics: Algebra 2 2007 Correlated to: Utah Core Curriculum for Math, Intermediate Algebra (Secondary)

Prentice Hall Mathematics: Algebra 2 2007 Correlated to: Utah Core Curriculum for Math, Intermediate Algebra (Secondary) Core Standards of the Course Standard 1 Students will acquire number sense and perform operations with real and complex numbers. Objective 1.1 Compute fluently and make reasonable estimates. 1. Simplify

More information

AP Calculus BC. All students enrolling in AP Calculus BC should have successfully completed AP Calculus AB.

AP Calculus BC. All students enrolling in AP Calculus BC should have successfully completed AP Calculus AB. AP Calculus BC Course Description: Advanced Placement Calculus BC is primarily concerned with developing the students understanding of the concepts of calculus and providing experiences with its methods

More information

Rational Expressions and Rational Equations

Rational Expressions and Rational Equations Rational Epressions and Rational Equations 6 6. Rational Epressions and Rational Functions 6. Multiplication and Division of Rational Epressions 6. Addition and Subtraction of Rational Epressions 6.4 Comple

More information

Trigonometry Lesson Objectives

Trigonometry Lesson Objectives Trigonometry Lesson Unit 1: RIGHT TRIANGLE TRIGONOMETRY Lengths of Sides Evaluate trigonometric expressions. Express trigonometric functions as ratios in terms of the sides of a right triangle. Use the

More information

Student Performance Q&A:

Student Performance Q&A: Student Performance Q&A: AP Calculus AB and Calculus BC Free-Response Questions The following comments on the free-response questions for AP Calculus AB and Calculus BC were written by the Chief Reader,

More information

ASA Angle Side Angle SAA Side Angle Angle SSA Side Side Angle. B a C

ASA Angle Side Angle SAA Side Angle Angle SSA Side Side Angle. B a C 8.2 The Law of Sines Section 8.2 Notes Page 1 The law of sines is used to solve for missing sides or angles of triangles when we have the following three cases: S ngle Side ngle S Side ngle ngle SS Side

More information

Math 2250 Exam #1 Practice Problem Solutions. g(x) = x., h(x) =

Math 2250 Exam #1 Practice Problem Solutions. g(x) = x., h(x) = Math 50 Eam # Practice Problem Solutions. Find the vertical asymptotes (if any) of the functions g() = + 4, h() = 4. Answer: The only number not in the domain of g is = 0, so the only place where g could

More information

9.3 OPERATIONS WITH RADICALS

9.3 OPERATIONS WITH RADICALS 9. Operations with Radicals (9 1) 87 9. OPERATIONS WITH RADICALS In this section Adding and Subtracting Radicals Multiplying Radicals Conjugates In this section we will use the ideas of Section 9.1 in

More information

DRAFT. Further mathematics. GCE AS and A level subject content

DRAFT. Further mathematics. GCE AS and A level subject content Further mathematics GCE AS and A level subject content July 2014 s Introduction Purpose Aims and objectives Subject content Structure Background knowledge Overarching themes Use of technology Detailed

More information

Answers to Basic Algebra Review

Answers to Basic Algebra Review Answers to Basic Algebra Review 1. -1.1 Follow the sign rules when adding and subtracting: If the numbers have the same sign, add them together and keep the sign. If the numbers have different signs, subtract

More information

This is a square root. The number under the radical is 9. (An asterisk * means multiply.)

This is a square root. The number under the radical is 9. (An asterisk * means multiply.) Page of Review of Radical Expressions and Equations Skills involving radicals can be divided into the following groups: Evaluate square roots or higher order roots. Simplify radical expressions. Rationalize

More information

THE SINE PRODUCT FORMULA AND THE GAMMA FUNCTION

THE SINE PRODUCT FORMULA AND THE GAMMA FUNCTION THE SINE PRODUCT FORMULA AND THE GAMMA FUNCTION ERICA CHAN DECEMBER 2, 2006 Abstract. The function sin is very important in mathematics and has many applications. In addition to its series epansion, it

More information

376 CURRICULUM AND SYLLABUS for Classes XI & XII

376 CURRICULUM AND SYLLABUS for Classes XI & XII 376 CURRICULUM AND SYLLABUS for Classes XI & XII MATHEMATICS CLASS - XI One Paper Time : 3 Hours 100 Marks Units Unitwise Weightage Marks Periods I. Sets Relations and Functions [9 marks] 1. Sets Relations

More information

Second Order Linear Nonhomogeneous Differential Equations; Method of Undetermined Coefficients. y + p(t) y + q(t) y = g(t), g(t) 0.

Second Order Linear Nonhomogeneous Differential Equations; Method of Undetermined Coefficients. y + p(t) y + q(t) y = g(t), g(t) 0. Second Order Linear Nonhomogeneous Differential Equations; Method of Undetermined Coefficients We will now turn our attention to nonhomogeneous second order linear equations, equations with the standard

More information

INVERSE TRIGONOMETRIC FUNCTIONS

INVERSE TRIGONOMETRIC FUNCTIONS Mathematics, in general, is fundamentally the science of self-evident things. FELIX KLEIN. Introduction In Chapter, we have studied that the inverse of a function f, denoted by f, eists if f is one-one

More information

Infinite Algebra 1 supports the teaching of the Common Core State Standards listed below.

Infinite Algebra 1 supports the teaching of the Common Core State Standards listed below. Infinite Algebra 1 Kuta Software LLC Common Core Alignment Software version 2.05 Last revised July 2015 Infinite Algebra 1 supports the teaching of the Common Core State Standards listed below. High School

More information

D.3. Angles and Degree Measure. Review of Trigonometric Functions

D.3. Angles and Degree Measure. Review of Trigonometric Functions APPENDIX D Precalculus Review D7 SECTION D. Review of Trigonometric Functions Angles and Degree Measure Radian Measure The Trigonometric Functions Evaluating Trigonometric Functions Solving Trigonometric

More information

MATH BOOK OF PROBLEMS SERIES. New from Pearson Custom Publishing!

MATH BOOK OF PROBLEMS SERIES. New from Pearson Custom Publishing! MATH BOOK OF PROBLEMS SERIES New from Pearson Custom Publishing! The Math Book of Problems Series is a database of math problems for the following courses: Pre-algebra Algebra Pre-calculus Calculus Statistics

More information

The numerical values that you find are called the solutions of the equation.

The numerical values that you find are called the solutions of the equation. Appendi F: Solving Equations The goal of solving equations When you are trying to solve an equation like: = 4, you are trying to determine all of the numerical values of that you could plug into that equation.

More information

Trigonometric Functions and Equations

Trigonometric Functions and Equations Contents Trigonometric Functions and Equations Lesson 1 Reasoning with Trigonometric Functions Investigations 1 Proving Trigonometric Identities... 271 2 Sum and Difference Identities... 276 3 Extending

More information

LIMITS AND CONTINUITY

LIMITS AND CONTINUITY LIMITS AND CONTINUITY 1 The concept of it Eample 11 Let f() = 2 4 Eamine the behavior of f() as approaches 2 2 Solution Let us compute some values of f() for close to 2, as in the tables below We see from

More information

Understanding Basic Calculus

Understanding Basic Calculus Understanding Basic Calculus S.K. Chung Dedicated to all the people who have helped me in my life. i Preface This book is a revised and expanded version of the lecture notes for Basic Calculus and other

More information

Mark Howell Gonzaga High School, Washington, D.C.

Mark Howell Gonzaga High School, Washington, D.C. Be Prepared for the Calculus Exam Mark Howell Gonzaga High School, Washington, D.C. Martha Montgomery Fremont City Schools, Fremont, Ohio Practice exam contributors: Benita Albert Oak Ridge High School,

More information

MATH ADVISEMENT GUIDE

MATH ADVISEMENT GUIDE MATH ADVISEMENT GUIDE Recommendations for math courses are based on your placement results, degree program and career interests. Placement score: MAT 001 or MAT 00 You must complete required mathematics

More information

Calculus I Notes. MATH /Richards/

Calculus I Notes. MATH /Richards/ Calculus I Notes MATH 52-154/Richards/10.1.07 The Derivative The derivative of a function f at p is denoted by f (p) and is informally defined by f (p)= the slope of the f-graph at p. Of course, the above

More information

Implicit Differentiation

Implicit Differentiation Revision Notes 2 Calculus 1270 Fall 2007 INSTRUCTOR: Peter Roper OFFICE: LCB 313 [EMAIL: roper@math.utah.edu] Standard Disclaimer These notes are not a complete review of the course thus far, and some

More information

1 TRIGONOMETRY. 1.0 Introduction. 1.1 Sum and product formulae. Objectives

1 TRIGONOMETRY. 1.0 Introduction. 1.1 Sum and product formulae. Objectives TRIGONOMETRY Chapter Trigonometry Objectives After studying this chapter you should be able to handle with confidence a wide range of trigonometric identities; be able to express linear combinations of

More information

The Derivative. Philippe B. Laval Kennesaw State University

The Derivative. Philippe B. Laval Kennesaw State University The Derivative Philippe B. Laval Kennesaw State University Abstract This handout is a summary of the material students should know regarding the definition and computation of the derivative 1 Definition

More information

Figure 1.1 Vector A and Vector F

Figure 1.1 Vector A and Vector F CHAPTER I VECTOR QUANTITIES Quantities are anything which can be measured, and stated with number. Quantities in physics are divided into two types; scalar and vector quantities. Scalar quantities have

More information

Paper II ( CALCULUS ) Shahada. College, Navapur. College, Shahada. Nandurbar

Paper II ( CALCULUS ) Shahada. College, Navapur. College, Shahada. Nandurbar Paper II ( CALCULUS ) Prof. R. B. Patel Dr. B. R. Ahirrao Prof. S. M. Patil Prof. A. S. Patil Prof. G. S. Patil Prof. A. D. Borse Art, Science & Comm. College, Shahada Jaihind College, Dhule Art, Science

More information

Mathematical Procedures

Mathematical Procedures CHAPTER 6 Mathematical Procedures 168 CHAPTER 6 Mathematical Procedures The multidisciplinary approach to medicine has incorporated a wide variety of mathematical procedures from the fields of physics,

More information

LESSON EIII.E EXPONENTS AND LOGARITHMS

LESSON EIII.E EXPONENTS AND LOGARITHMS LESSON EIII.E EXPONENTS AND LOGARITHMS LESSON EIII.E EXPONENTS AND LOGARITHMS OVERVIEW Here s what ou ll learn in this lesson: Eponential Functions a. Graphing eponential functions b. Applications of eponential

More information

1 Lecture 19: Implicit differentiation

1 Lecture 19: Implicit differentiation Lecture 9: Implicit differentiation. Outline The technique of implicit differentiation Tangent lines to a circle Examples.2 Implicit differentiation Suppose we have two quantities or variables x and y

More information

PRINCIPLES OF PROBLEM SOLVING

PRINCIPLES OF PROBLEM SOLVING PRINCIPLES OF PROBLEM SOLVING There are no hard and fast rules that will ensure success in solving problems. However, it is possible to outline some general steps in the problem-solving process and to

More information

4.1 Radian and Degree Measure

4.1 Radian and Degree Measure Date: 4.1 Radian and Degree Measure Syllabus Objective: 3.1 The student will solve problems using the unit circle. Trigonometry means the measure of triangles. Terminal side Initial side Standard Position

More information

Students Currently in Algebra 2 Maine East Math Placement Exam Review Problems

Students Currently in Algebra 2 Maine East Math Placement Exam Review Problems Students Currently in Algebra Maine East Math Placement Eam Review Problems The actual placement eam has 100 questions 3 hours. The placement eam is free response students must solve questions and write

More information

Polynomial Degree and Finite Differences

Polynomial Degree and Finite Differences CONDENSED LESSON 7.1 Polynomial Degree and Finite Differences In this lesson you will learn the terminology associated with polynomials use the finite differences method to determine the degree of a polynomial

More information

PRE-CALCULUS GRADE 12

PRE-CALCULUS GRADE 12 PRE-CALCULUS GRADE 12 [C] Communication Trigonometry General Outcome: Develop trigonometric reasoning. A1. Demonstrate an understanding of angles in standard position, expressed in degrees and radians.

More information

Definition of derivative

Definition of derivative Definition of derivative Contents 1. Slope-The Concept 2. Slope of a curve 3. Derivative-The Concept 4. Illustration of Example 5. Definition of Derivative 6. Example 7. Extension of the idea 8. Example

More information

LINEAR INEQUALITIES. less than, < 2x + 5 x 3 less than or equal to, greater than, > 3x 2 x 6 greater than or equal to,

LINEAR INEQUALITIES. less than, < 2x + 5 x 3 less than or equal to, greater than, > 3x 2 x 6 greater than or equal to, LINEAR INEQUALITIES When we use the equal sign in an equation we are stating that both sides of the equation are equal to each other. In an inequality, we are stating that both sides of the equation are

More information

PYTHAGOREAN TRIPLES KEITH CONRAD

PYTHAGOREAN TRIPLES KEITH CONRAD PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient

More information

Arithmetic Operations. The real numbers have the following properties: In particular, putting a 1 in the Distributive Law, we get

Arithmetic Operations. The real numbers have the following properties: In particular, putting a 1 in the Distributive Law, we get Review of Algebra REVIEW OF ALGEBRA Review of Algebra Here we review the basic rules and procedures of algebra that you need to know in order to be successful in calculus. Arithmetic Operations The real

More information

WASSCE / WAEC ELECTIVE / FURTHER MATHEMATICS SYLLABUS

WASSCE / WAEC ELECTIVE / FURTHER MATHEMATICS SYLLABUS Visit this link to read the introductory text for this syllabus. 1. Circular Measure Lengths of Arcs of circles and Radians Perimeters of Sectors and Segments measure in radians 2. Trigonometry (i) Sine,

More information

TSI College Level Math Practice Test

TSI College Level Math Practice Test TSI College Level Math Practice Test Tutorial Services Mission del Paso Campus. Factor the Following Polynomials 4 a. 6 8 b. c. 7 d. ab + a + b + 6 e. 9 f. 6 9. Perform the indicated operation a. ( +7y)

More information

Answer Key for California State Standards: Algebra I

Answer Key for California State Standards: Algebra I Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.

More information

x(x + 5) x 2 25 (x + 5)(x 5) = x 6(x 4) x ( x 4) + 3

x(x + 5) x 2 25 (x + 5)(x 5) = x 6(x 4) x ( x 4) + 3 CORE 4 Summary Notes Rational Expressions Factorise all expressions where possible Cancel any factors common to the numerator and denominator x + 5x x(x + 5) x 5 (x + 5)(x 5) x x 5 To add or subtract -

More information

Math 2443, Section 16.3

Math 2443, Section 16.3 Math 44, Section 6. Review These notes will supplement not replace) the lectures based on Section 6. Section 6. i) ouble integrals over general regions: We defined double integrals over rectangles in the

More information