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) =

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "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) ="

Transcription

1 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 f (x) = lim x a Find the vertical asymptotes of f (x) = x 3 x 2 9. Solution: lim f (x) = x a + f (x) = x a + Candidates to consider are points that give division by zero: x = ±3. x 3 We can use a word argument to deduce lim x 3 x 2 9 =. Thus, x = 3 is a vertical asymptote. On the other hand, x = 3 is NOT a vertical asymptote since the limit is: x 3 lim x 3 x 2 9 = lim 1 x 3 x + 3 = 1 6 and is not equal to ±. x = 3 is a hole in the function since it cancels. AMAT 217 (University of Calgary) Fall / 24

2 Horizontal asymptotes Definition of Horizontal Asymptote The line y = L is a horizontal asymptote of f (x) if L is finite and either lim f (x) = L or lim f (x) = L. x x Find the horizontal asymptotes of f (x) = x x. Solution: We must compute two limits: x and x. x. Notice that for x arbitrarily large that x > 0 so that x = x. In particular, for x in the interval (0, ) we have lim x x. x x = lim x x x = 1. Notice that for x arbitrarily large negative that x < 0 so that x = x. x For x in the interval (, 0) we have lim x x = lim x x x = 1. Therefore there are two horizontal asymptotes, namely, y = 1 and y = 1. AMAT 217 (University of Calgary) Fall / 24

3 Derivative Rules Know these! (c) = 0 (ax) = a (x n ) = nx n 1 (cf ) = cf (f ± g) = f ± g (fg) = f g + fg ( ) f = f g fg g g 2 [f (g(x))] = f (g(x)) g (x) (e x ) = e x (ln x) = 1 x (sin x) = cos x (cos x) = sin x (tan x) = sec 2 x (function) (function) Use logarithmic differentiation Inverse Trig Derivatives ( sin 1 x ) = 1 1 x 2 ( cos 1 x ) = 1 1 x 2 ( tan 1 x ) = x 2 Hyperbolic Derivatives (sinh x) = cosh x (cosh x) = sinh x (tanh x) = sech 2 x AMAT 217 (University of Calgary) Fall / 24

4 Find constants a, b so that f (x) = { ax 2 + 2x if x < 2 4b if x 2 is differentiable at x = 2. For differentiability at x = 2 we need f (x) to: (i) be continuous at x = 2, [Recall: differentiable continuous but continuous differentiable] (ii) have matching right and left derivatives at x = 2. The first condition requires that left and right hand limits be equal: lim f (x) = x 2 lim 2 x 2 (ax + 2x) = 4a + 4 and lim f (x) = x 2 + Thus, we require 4a + 4 = 4b, that is, a + 1 = b. lim = 4b x 2 +(4b) The second condition requires that left and right derivatives are equal: f (2) = d ( ax 2 + 2x) = (2ax + 2) dx = 4a + 2 and f +(2) = d x=2 dx x=2 (4b) = 0 x=2 Thus, we require 4a + 2 = 0, that is, 2a + 1 = 0. We have two equations and solving gives a = 1/2 and b = 1/2. Therefore, for f (x) to be differentiable at x = 2 we require a = 1/2 and b = 1/2. AMAT 217 (University of Calgary) Fall / 24

5 (From past exam/midterm) { 1 4x if x < 1 Suppose f (x) = x if x 1 Is f (x) differentiable at x = 1? If so, find f ( 1). Solution: First, we need to verify f (x) is continuous at x = 1. Thus, we need left and right hand limits be equal: lim f (x) = x 1 lim 4x) = 3 and lim x 1 ( 1 f (x) = x 1 + Thus, f (x) is continuous at x = 1 since f ( 1) = lim f (x) = 3. x 1 Now we need left and right hand derivatives be equal: From the left of x = 1, the derivative is: f ( 1) = d dx ( 1 4x) x= 1 = 4 lim + 2) = 3 x 1 +(x4 From the right of x = 1, the derivative is: f + ( 1) = d ( x ) dx x= 1 = 4x 3 x= 1 = 4( 1) 3 = 4 Since continuous and left/right derivatives match, we have f ( 1) = 4. AMAT 217 (University of Calgary) Fall / 24

6 Implicit Differentiation (with respect to x) 1 Differentiate both sides as usual, 2 Whenever you do the derivative of terms with y, multiply by dy dx. Find the equation of the tangent line to tan(x + y) = x + 1 at P = (0, π/4). Recall: The slope of the tangent line is the derivative dy dx P=(0,π/4). Differentiate both sides with respect to x thinking of y as a function of x: ( sec 2 (x + y) 1 + dy ) = dx } {{ } derivative of x + y The left side used the chain rule combined with (tan x) = sec 2 x. We have x = 0, y = π/4 and m = dy dx P=(0,π/4), so: sec 2 (0 + π/4)(1 + m) = 1 1 Since sec(π/4) = cos(π/4) = 1 1/ 2 = 2, we have: 2(1 + m) = m = 1 m = Thus, the equation of the tangent line is then (y π/4) = ( 1/2)(x 0). AMAT 217 (University of Calgary) Fall / 24

7 Logarithmic Differentiation 1 Take ln of both sides of y = f (x) to get ln y = ln f (x). 2 Simplify using log properties. 3 Differentiate with respect to x using (ln y) = y y on left side. 4 Multiply by y and replace y with original function y = f (x). Find h (x) given h(x) = (x 2 + 1) 2x. We take the natural log of both sides: ln h(x) = ln((x 2 + 1) 2x ) Using log properties we get: ln h(x) = 2x ln(x 2 + 1) Taking the derivative gives: h (x) h(x) = 2 2x ln(x2 + 1) + 2x x Solving for h (x) and replacing h(x) = (x 2 + ( 1) 2x gives: ) h (x) = (x 2 + 1) 2x 2 ln(x 2 + 1) + 4x2 x AMAT 217 (University of Calgary) Fall / 24

8 By comparing powers we see that 3 = 300/T so that T = 100. AMAT 217 (University of Calgary) Fall / 24 Application: Half-life The half-life formula is ) t/t y(t) = y 0 ( 1 2 y(t) = amount at time t, y 0 = original amount, t = time (in years), T = half-life (in years). Find the half-life of a radioactive substance if after 300 years only 12.5% of the original amount remains. We are given that y(300) is 12.5% of the original amount y 0. That is, y(300) = y 0. Thus, substituting t = 300 into the formula gives: ( ) 1 300/T y 0 = y 0 2 Cancelling y 0 s gives: ( ) 1 300/T = 2 ( ) 1 = 8 ( ) 1 300/T 2 ( ) 1 3 = 2 ( ) 1 300/T 2

9 Application: Newton s Law of Cooling The Newton s Law of Cooling formula is T (t) = T e + (T 0 T e)e kt T (t) = temperature of object at time t, T 0 = original temperature of object, T e = temperature of environment (medium/room temp), k = a constant dependent on the material properties of the object. An object of temperature 60 C is placed in a room with temperature 20 C. At the end of 5 minutes the object has cooled to a temperature of 40 C. What is the temperature of the object at the end of 15 minutes? We are given that T 0 = 60 and T e = 20: T (t) = e kt We are also given that T (5) = 40 which will allow us to find k: We need to find T (15): 40 = e k(5) 20 = 40e 5k k = ln(1/2) 5 T (15) = e ln(1/2) (15) 5 = e ln(1/2)3 = Therefore, the temperature at the end of 15 minutes is 25 C. ( ) 1 3 = 25 2 AMAT 217 (University of Calgary) Fall / 24

10 Application: Related Rates (Related Rates #12 on Worksheet) When air expands isothermally (i.e., expands at constant temperature), the pressure P and volume V are related by Boyle s Law: PV = C, where C is a constant. At a certain instant, the pressure is 40 pounds per square inch, the volume is 8 cubic feet and is increasing at the rate of 0.5 cubic feet per second. At what rate is the pressure changing? We are given: P = 40 psi V = 8 ft 3 dv dt = 0.5 ft3 /s We need to compute dp dt. Since we are already given the relationship between P and V we differentiate PV = C with respect to time (using the product rule on PV ): Plugging in the given information gives: dp dt V + P dv dt = 0 dp dt (8) + (40)(0.5) = 0 dp = 2.5 psi/s dt Thus, the pressure is decreasing by 2.5 pounds per square inch per second. AMAT 217 (University of Calgary) Fall / 24

11 Application: Newton s Method Newton s Method Goal: To approximate a root of f (x) = 0 with an initial guess of x = x 0. If x n is the current approximation, then the next approximation is: x n+1 = x n f (xn) f (x n) Use Newton s method with an initial approximation of x 0 = 1 to find the approximation x 2 of the equation x 3 4 = 0 by taking f (x) = x 3 4. If x is the current approximation, the next approximation is given by: g(x) = x f (xn) f (x = x x3 4 n) 3x 2 = 2x x 2 Given that x 0 = 1, we compute x 1 by plugging x 0 = 1 into g(x): x 1 = g(x 0 ) = g(1) = = 2 3 As x 1 = 2, we compute x 2 by plugging x 1 = 2 into g(x): x 2 = g(x 1 ) = g(2) = = Thus, the approximation x 2 is equal to 5/3. AMAT 217 (University of Calgary) Fall / 24

12 Application: Linearizations The linearization of a function is the equation of its tangent line. Linearization of f(x) at x=a L(x) = f (a) + f (a)(x a). Find the linearization of f (x) = tan(sin(x 1)) 4x + 1 at the point on the curve where x = 1. Recall that the slope of the tangent line at x = 1 is equal to f (1). We have f (x) = sec 2 (sin(x 1)) cos(x 1) 1 4 Thus, when x = 1 we have: f (1) = sec 2 (sin(0)) cos(0) 4 Since sin 0 = 0, cos 0 = 1 and sec 0 = 1 cos 0 = 1 1 = 1: f (1) = sec 2 (0) 4 = 1 4 = 3 Now we use the point-slope formula for a straight line: y y 1 = m(x x 1 ). The slope is m = 3, and x 1 = 1. To get y 1 we sub x = 1 into f (x): f (1) = tan(sin(0)) 4(1) + 1 = 3 Thus, the equation of the tangent line at x = 1 is: y ( 3) = ( 3)(x 1) L(x) = 3x AMAT 217 (University of Calgary) Fall / 24

13 Geometry Some application problems require geometry, so it helps to go over the basics: Pythagorean Theorem know how to get equations using similar triangles circumference of a circle of radius r is C = 2πr area of a rectangle is length times width (A = lw) area of a circle of radius r is A = πr 2 area of a triangle with base length b and height h is A = bh 2 volume of a rectangular box is length times width times height (V = lwh) volume of a cylinder is V = πr 2 h volume of a right circular cone with base radius r and height h is V = 1 3 πr 2 h surface area of a box: you must add up the areas of six rectangles surface area of a cylinder: you must add up the areas of two circles and one rectangle perimeter: you must add up the length of a bunch of line segments trigonometric ratios and SOH CAH TOA cost associated with length: (cost of material) times (length) cost associated with area: (cost of material) times (area) : If building a rectangular fence with front fencing costing $4/m while sides and back fencing is $2.50/m (since you want the front to look nice and use more expensive material), the cost function is then: C = 4(x) + 2.5(y + x + y) where x is length of front and back, y is length of left and right sides. AMAT 217 (University of Calgary) Fall / 24

14 Absolute Extreme Points To find absolute extrema of a continuous function f (x) on a closed interval [a, b], plug in critical points, singular points and endpoints into f (x) (that are in the interval) to get the highest and lowest points of the function. Find the absolute extrema of f (x) = x on the interval [ 1, 2]. The derivative is: f (x) = 2x x = 0 is a critical point (there are no singular points) Plugging in endpoints and critical points belonging to [ 1, 2] gives: Critical Points: f (0) = = 1 Singular Points: None Endpoints: f ( 1) = 2 f (2) = 5 Comparing gives: The absolute maximum of f (x) is 5 which occurs at x = 2 The absolute minimum is 1 which occurs at x = 0 AMAT 217 (University of Calgary) Fall / 24

15 Local Extreme Points To find local extrema of f (x), find the critical points and singular points and classify them using the first derivative test. The Increasing and Decreasing Test 1 If f (x) > 0 on some interval I, then f (x) is increasing on I. 2 If f (x) < 0 on some interval I, then f (x) is decreasing on I. Determine local extrema of f (x) = x 3 + 6x 2 15x + 7. Solution:The derivative is: f (x) = 3(x 1)(x + 5), with critical points x = 5, 1 and intervals of increasing/decreasing as illustrated: Therefore, x = 1 is a local minimum and x = 5 is a local maximum. AMAT 217 (University of Calgary) Fall / 24

16 Concavity Test If f (x) > 0 on some interval I, then f (x) is concave up on I. If f (x) < 0 on some interval I, then f (x) is concave down on I. Possible inflection points are where f (x) = 0 or DNE (check the concavity on both sides). Identify the intervals of concavity and inflection points for f (x) = x 3 + 6x 2 15x + 7. The process for determining concavity is almost identical to that of determining increasing/decreasing (except we use the second derivative instead). The second derivative is: f (x) = 6(x + 2). We first determine possible inflection points (where f (x) = 0 or f (x) does not exist). In our case, f (x) = 0 when x = 2. Draw a number line and sub test points into f (x): f (x) is concave down on (, 2) and concave up on ( 2, ). Since concavity changes at x = 2, then x = 2 is an inflection point. AMAT 217 (University of Calgary) Fall / 24

17 Anti-derivatives Integral Rules Power Rules: Trig Rules: Hyperbolic: Constant Multiple Rule: Sum/Difference Rule: Inverse Trig Rules: Constant Rule: x n dx = xn+1 n C, Exponent Rule: sin x dx = cos x + C. sinh x dx = cosh x + C. kf (x) dx = k k dx = kx + C. f (x) ± g(x) dx = (n 1), x 2 dx = tan 1 x + C f (x) dx, e x dx = e x + C. cos x dx = sin x + C. cosh x dx = sinh x + C. f (x) dx ± k is constant. g(x) dx. 1 dx = ln x + C, (x 0). x sec 2 x dx = tan x + C. sech 2 x dx = tanh x + C. 1 1 x 2 dx = sin 1 x + C. AMAT 217 (University of Calgary) Fall / 24

18 How to evaluate integrals? Integration Techniques 1 The basics (a) Use a formula (b) Expand products (c) Split up fractions (d) Complete the square 2 Substitution Rule Evaluate (z + 3 z)(4 z 2 ) dz Solution: Remember that we don t have a rule to deal with integrals of products. In this case we expand products: (z + 3 z)(4 z 2 ) dz = 4z z 3 + 4z 1/3 z 7/3 dz ( z 2 = 4 2 ) ( z z 4/3 4/3 ) = 2z 2 z z4/3 3z10/3 + C 10 z10/3 10/3 + C AMAT 217 (University of Calgary) Fall / 24

19 Substitution Rule: If u = g(x) is a differentiable function whose range is an interval I and f is continuous on I, then f (g(x))g (x) dx = f (u) du. A general strategy to follow is as follows. Substitution Rule Strategy 1 Choose a possible u = u(x), try one that appears inside a function. 2 Calculate du = u (x) dx. 3 Either replace u (x) dx by du, or replace dx by 4 Write the rest of the integrand in terms of u. 5 Integrate. 6 Rewrite the result back in terms of x. du u, and cancel. (x) AMAT 217 (University of Calgary) Fall / 24

20 Evaluate sec 4 x tan 6 x dx We try u = tan x, so that du = sec 2 x dx, i.e., dx = du sec 2 x : sec 4 x tan 6 x dx = = = = = sec 4 x (u 6 du ) sec 2 Using the substitution x sec 2 x(u 6 ) du Cancelling (1 + tan 2 x)(u 6 ) du Using sec 2 x = 1 + tan 2 x (1 + u 2 )(u 6 ) du Using the substitution u = tan x (u 6 + u 8 ) du Expanding. = u7 7 + u9 9 + C Since = tan7 x 7 + tan9 x 9 x n dx = xn+1 + C, n 1 n C Replacing u back in terms of x AMAT 217 (University of Calgary) Fall / 24

21 Definite Integrals - Areas 1 Compute x dx. 1 b f (x) dx = Net Area under f (x) from x = a to x = b a Recall that y = x forms a V -shape when graphing it. The integral is equal to the net area under the V from x = 1 to x = 1. This gives two triangles, each with base of 1 and height of 1. 1 The net area is then 1, thus, x dx = Evaluate the following integral x dx and hence, computing the area under y = x from x = 0 0 to x = 1 (the area of a triangle with base and width 1). 1 0 x dx = x2 2 1 = AMAT 217 (University of Calgary) Fall / 24

22 The Fundamental Theorem of Calculus - Part I FTC I + Chain Rule: d dx v(x) f (t) dt = f (v(x))v (x) f (u(x))u (x) u(x) Differentiate the following integral with respect to x: Solution: x 2 t 3 sin(1 + t) dt. 10x We have f (t) = t 3 sin(1 + t), u(x) = 10x and v(x) = x 2. Then u (x) = 10 and v (x) = 2x. Thus, d x 2 t 3 sin(1 + t) dt dx 10x = [ (x 2 ) 3 sin(1 + (x 2 )) ] [2x] [ (10x) 3 sin(1 + (10x)) ] [10] = 2x 7 sin(1 + x 2 ) 10000x 3 sin(1 + 10x) AMAT 217 (University of Calgary) Fall / 24

23 Area Between Two Curves Problem: Given two curves y = f (x) and y = g(x), determine the area enclosed between them. Solution: The area A of the region bounded by the curves y = f (x) and y = g(x) and the lines x = a and x = b is: b A = f (x) g(x) dx. a Informally this can be thought of as follows: b Area = (top curve) (bottom curve) dx, a x b. a AMAT 217 (University of Calgary) Fall / 24

24 Determine the area enclosed by y = x 2, y = x, x = 0 and x = 2. Solution: From the last problem we have the following sketch: Since the top curve changes at x = 1, we need to use the formula twice. For A 1 we have a = 0, b = 1, the top curve is y = x and the bottom curve is y = x 2. For A 2 we have a = 1, b = 2, the top curve is y = x 2 and the bottom curve is y = x. Area = A1 + A2 = 1 ( 2 x x 2 ) dx + (x 2 x) dx = AMAT 217 (University of Calgary) Fall / 24

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

MATH SOLUTIONS TO PRACTICE FINAL EXAM. (x 2)(x + 2) (x 2)(x 3) = x + 2. x 2 x 2 5x + 6 = = 4.

MATH SOLUTIONS TO PRACTICE FINAL EXAM. (x 2)(x + 2) (x 2)(x 3) = x + 2. x 2 x 2 5x + 6 = = 4. MATH 55 SOLUTIONS TO PRACTICE FINAL EXAM x 2 4.Compute x 2 x 2 5x + 6. When x 2, So x 2 4 x 2 5x + 6 = (x 2)(x + 2) (x 2)(x 3) = x + 2 x 3. x 2 4 x 2 x 2 5x + 6 = 2 + 2 2 3 = 4. x 2 9 2. Compute x + sin

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

Math 120 Final Exam Practice Problems, Form: A

Math 120 Final Exam Practice Problems, Form: A Math 120 Final Exam Practice Problems, Form: A Name: While every attempt was made to be complete in the types of problems given below, we make no guarantees about the completeness of the problems. Specifically,

More information

Derivatives: rules and applications (Stewart Ch. 3/4) The derivative f (x) of the function f(x):

Derivatives: rules and applications (Stewart Ch. 3/4) The derivative f (x) of the function f(x): Derivatives: rules and applications (Stewart Ch. 3/4) The derivative f (x) of the function f(x): f f(x + h) f(x) (x) = lim h 0 h (for all x for which f is differentiable/ the limit exists) Property:if

More information

PRACTICE FINAL. Problem 1. Find the dimensions of the isosceles triangle with largest area that can be inscribed in a circle of radius 10cm.

PRACTICE FINAL. Problem 1. Find the dimensions of the isosceles triangle with largest area that can be inscribed in a circle of radius 10cm. PRACTICE FINAL Problem 1. Find the dimensions of the isosceles triangle with largest area that can be inscribed in a circle of radius 1cm. Solution. Let x be the distance between the center of the circle

More information

Free Response Questions Compiled by Kaye Autrey for face-to-face student instruction in the AP Calculus classroom

Free Response Questions Compiled by Kaye Autrey for face-to-face student instruction in the AP Calculus classroom Free Response Questions 1969-005 Compiled by Kaye Autrey for face-to-face student instruction in the AP Calculus classroom 1 AP Calculus Free-Response Questions 1969 AB 1 Consider the following functions

More information

f (x) has an absolute minimum value f (c) at the point x = c in its domain if

f (x) has an absolute minimum value f (c) at the point x = c in its domain if Definitions - Absolute maximum and minimum values f (x) has an absolute maximum value f (c) at the point x = c in its domain if f (x) f (c) holds for every x in the domain of f (x). f (x) has an absolute

More information

Calculus 1: Sample Questions, Final Exam, Solutions

Calculus 1: Sample Questions, Final Exam, Solutions Calculus : Sample Questions, Final Exam, Solutions. Short answer. Put your answer in the blank. NO PARTIAL CREDIT! (a) (b) (c) (d) (e) e 3 e Evaluate dx. Your answer should be in the x form of an integer.

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 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

Apr 23, 2015. Calculus with Algebra and Trigonometry II Lecture 23Final Review: Apr Curve 23, 2015 sketching 1 / and 19pa

Apr 23, 2015. Calculus with Algebra and Trigonometry II Lecture 23Final Review: Apr Curve 23, 2015 sketching 1 / and 19pa Calculus with Algebra and Trigonometry II Lecture 23 Final Review: Curve sketching and parametric equations Apr 23, 2015 Calculus with Algebra and Trigonometry II Lecture 23Final Review: Apr Curve 23,

More information

5 Indefinite integral

5 Indefinite integral 5 Indefinite integral The most of the mathematical operations have inverse operations: the inverse operation of addition is subtraction, the inverse operation of multiplication is division, the inverse

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

CHAPTER 13. Definite Integrals. Since integration can be used in a practical sense in many applications it is often

CHAPTER 13. Definite Integrals. Since integration can be used in a practical sense in many applications it is often 7 CHAPTER Definite Integrals Since integration can be used in a practical sense in many applications it is often useful to have integrals evaluated for different values of the variable of integration.

More information

Section 2.1 Rectangular Coordinate Systems

Section 2.1 Rectangular Coordinate Systems P a g e 1 Section 2.1 Rectangular Coordinate Systems 1. Pythagorean Theorem In a right triangle, the lengths of the sides are related by the equation where a and b are the lengths of the legs and c is

More information

PROBLEM SET. Practice Problems for Exam #2. Math 2350, Fall Nov. 7, 2004 Corrected Nov. 10 ANSWERS

PROBLEM SET. Practice Problems for Exam #2. Math 2350, Fall Nov. 7, 2004 Corrected Nov. 10 ANSWERS PROBLEM SET Practice Problems for Exam #2 Math 2350, Fall 2004 Nov. 7, 2004 Corrected Nov. 10 ANSWERS i Problem 1. Consider the function f(x, y) = xy 2 sin(x 2 y). Find the partial derivatives f x, f y,

More information

a b c d e You have two hours to do this exam. Please write your name on this page, and at the top of page three. GOOD LUCK! 3. a b c d e 12.

a b c d e You have two hours to do this exam. Please write your name on this page, and at the top of page three. GOOD LUCK! 3. a b c d e 12. MA123 Elem. Calculus Fall 2015 Exam 2 2015-10-22 Name: Sec.: Do not remove this answer page you will turn in the entire exam. No books or notes may be used. You may use an ACT-approved calculator during

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

SAT Subject Math Level 2 Facts & Formulas

SAT Subject Math Level 2 Facts & Formulas Numbers, Sequences, Factors Integers:..., -3, -2, -1, 0, 1, 2, 3,... Reals: integers plus fractions, decimals, and irrationals ( 2, 3, π, etc.) Order Of Operations: Arithmetic Sequences: PEMDAS (Parentheses

More information

5.1 Derivatives and Graphs

5.1 Derivatives and Graphs 5.1 Derivatives and Graphs What does f say about f? If f (x) > 0 on an interval, then f is INCREASING on that interval. If f (x) < 0 on an interval, then f is DECREASING on that interval. A function has

More information

Calculus 1st Semester Final Review

Calculus 1st Semester Final Review Calculus st Semester Final Review Use the graph to find lim f ( ) (if it eists) 0 9 Determine the value of c so that f() is continuous on the entire real line if f ( ) R S T, c /, > 0 Find the limit: lim

More information

MTH 233 Calculus I. Part 1: Introduction, Limits and Continuity

MTH 233 Calculus I. Part 1: Introduction, Limits and Continuity MTH 233 Calculus I Tan, Single Variable Calculus: Early Transcendentals, 1st ed. Part 1: Introduction, Limits and Continuity 0 Preliminaries It is assumed that students are comfortable with the material

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

4 More Applications of Definite Integrals: Volumes, arclength and other matters

4 More Applications of Definite Integrals: Volumes, arclength and other matters 4 More Applications of Definite Integrals: Volumes, arclength and other matters Volumes of surfaces of revolution 4. Find the volume of a cone whose height h is equal to its base radius r, by using the

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

MATH 121 FINAL EXAM FALL 2010-2011. December 6, 2010

MATH 121 FINAL EXAM FALL 2010-2011. December 6, 2010 MATH 11 FINAL EXAM FALL 010-011 December 6, 010 NAME: SECTION: Instructions: Show all work and mark your answers clearly to receive full credit. This is a closed notes, closed book exam. No electronic

More information

D f = (2, ) (x + 1)(x 3) (b) g(x) = x 1 solution: We need the thing inside the root to be greater than or equal to 0. So we set up a sign table.

D f = (2, ) (x + 1)(x 3) (b) g(x) = x 1 solution: We need the thing inside the root to be greater than or equal to 0. So we set up a sign table. . Find the domains of the following functions: (a) f(x) = ln(x ) We need x > 0, or x >. Thus D f = (, ) (x + )(x 3) (b) g(x) = x We need the thing inside the root to be greater than or equal to 0. So we

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

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

TOPIC 3: CONTINUITY OF FUNCTIONS

TOPIC 3: CONTINUITY OF FUNCTIONS TOPIC 3: CONTINUITY OF FUNCTIONS. Absolute value We work in the field of real numbers, R. For the study of the properties of functions we need the concept of absolute value of a number. Definition.. Let

More information

Math 113 HW #7 Solutions

Math 113 HW #7 Solutions Math 3 HW #7 Solutions 35 0 Given find /dx by implicit differentiation y 5 + x 2 y 3 = + ye x2 Answer: Differentiating both sides with respect to x yields 5y 4 dx + 2xy3 + x 2 3y 2 ) dx = dx ex2 + y2x)e

More information

Pre-Calculus Review Lesson 1 Polynomials and Rational Functions

Pre-Calculus Review Lesson 1 Polynomials and Rational Functions If a and b are real numbers and a < b, then Pre-Calculus Review Lesson 1 Polynomials and Rational Functions For any real number c, a + c < b + c. For any real numbers c and d, if c < d, then a + c < b

More information

Solutions to Homework 10

Solutions to Homework 10 Solutions to Homework 1 Section 7., exercise # 1 (b,d): (b) Compute the value of R f dv, where f(x, y) = y/x and R = [1, 3] [, 4]. Solution: Since f is continuous over R, f is integrable over R. Let x

More information

MA107 Precalculus Algebra Exam 2 Review Solutions

MA107 Precalculus Algebra Exam 2 Review Solutions MA107 Precalculus Algebra Exam 2 Review Solutions February 24, 2008 1. The following demand equation models the number of units sold, x, of a product as a function of price, p. x = 4p + 200 a. Please write

More information

21-114: Calculus for Architecture Homework #1 Solutions

21-114: Calculus for Architecture Homework #1 Solutions 21-114: Calculus for Architecture Homework #1 Solutions November 9, 2004 Mike Picollelli 1.1 #26. Find the domain of g(u) = u + 4 u. Solution: We solve this by considering the terms in the sum separately:

More information

2 Integrating Both Sides

2 Integrating Both Sides 2 Integrating Both Sides So far, the only general method we have for solving differential equations involves equations of the form y = f(x), where f(x) is any function of x. The solution to such an equation

More information

Applications of Integration Day 1

Applications of Integration Day 1 Applications of Integration Day 1 Area Under Curves and Between Curves Example 1 Find the area under the curve y = x2 from x = 1 to x = 5. (What does it mean to take a slice?) Example 2 Find the area under

More information

AP Calculus AB First Semester Final Exam Practice Test Content covers chapters 1-3 Name: Date: Period:

AP Calculus AB First Semester Final Exam Practice Test Content covers chapters 1-3 Name: Date: Period: AP Calculus AB First Semester Final Eam Practice Test Content covers chapters 1- Name: Date: Period: This is a big tamale review for the final eam. Of the 69 questions on this review, questions will be

More information

55x 3 + 23, f(x) = x2 3. x x 2x + 3 = lim (1 x 4 )/x x (2x + 3)/x = lim

55x 3 + 23, f(x) = x2 3. x x 2x + 3 = lim (1 x 4 )/x x (2x + 3)/x = lim Slant Asymptotes If lim x [f(x) (ax + b)] = 0 or lim x [f(x) (ax + b)] = 0, then the line y = ax + b is a slant asymptote to the graph y = f(x). If lim x f(x) (ax + b) = 0, this means that the graph of

More information

Lecture 3 : The Natural Exponential Function: f(x) = exp(x) = e x. y = exp(x) if and only if x = ln(y)

Lecture 3 : The Natural Exponential Function: f(x) = exp(x) = e x. y = exp(x) if and only if x = ln(y) Lecture 3 : The Natural Exponential Function: f(x) = exp(x) = Last day, we saw that the function f(x) = ln x is one-to-one, with domain (, ) and range (, ). We can conclude that f(x) has an inverse function

More information

Review for Calculus Rational Functions, Logarithms & Exponentials

Review for Calculus Rational Functions, Logarithms & Exponentials Definition and Domain of Rational Functions A rational function is defined as the quotient of two polynomial functions. F(x) = P(x) / Q(x) The domain of F is the set of all real numbers except those for

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

Math 41: Calculus Final Exam December 7, 2009

Math 41: Calculus Final Exam December 7, 2009 Math 41: Calculus Final Exam December 7, 2009 Name: SUID#: Select your section: Atoshi Chowdhury Yuncheng Lin Ian Petrow Ha Pham Yu-jong Tzeng 02 (11-11:50am) 08 (10-10:50am) 04 (1:15-2:05pm) 03 (11-11:50am)

More information

Practice Problems for Midterm 2

Practice Problems for Midterm 2 Practice Problems for Midterm () For each of the following, find and sketch the domain, find the range (unless otherwise indicated), and evaluate the function at the given point P : (a) f(x, y) = + 4 y,

More information

Calculus AB 2014 Scoring Guidelines

Calculus AB 2014 Scoring Guidelines P Calculus B 014 Scoring Guidelines 014 The College Board. College Board, dvanced Placement Program, P, P Central, and the acorn logo are registered trademarks of the College Board. P Central is the official

More information

Math 1B, lecture 5: area and volume

Math 1B, lecture 5: area and volume Math B, lecture 5: area and volume Nathan Pflueger 6 September 2 Introduction This lecture and the next will be concerned with the computation of areas of regions in the plane, and volumes of regions in

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

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

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

APPLICATIONS OF DIFFERENTIATION

APPLICATIONS OF DIFFERENTIATION 4 APPLICATIONS OF DIFFERENTIATION APPLICATIONS OF DIFFERENTIATION So far, we have been concerned with some particular aspects of curve sketching: Domain, range, and symmetry (Chapter 1) Limits, continuity,

More information

Math 181 Spring 2007 HW 1 Corrected

Math 181 Spring 2007 HW 1 Corrected Math 181 Spring 2007 HW 1 Corrected February 1, 2007 Sec. 1.1 # 2 The graphs of f and g are given (see the graph in the book). (a) State the values of f( 4) and g(3). Find 4 on the x-axis (horizontal axis)

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

6 Further differentiation and integration techniques

6 Further differentiation and integration techniques 56 6 Further differentiation and integration techniques Here are three more rules for differentiation and two more integration techniques. 6.1 The product rule for differentiation Textbook: Section 2.7

More information

Curriculum Map. Discipline: Math Course: AP Calculus AB Teacher: Louis Beuschlein

Curriculum Map. Discipline: Math Course: AP Calculus AB Teacher: Louis Beuschlein Curriculum Map Discipline: Math Course: AP Calculus AB Teacher: Louis Beuschlein August/September: State: 8.B.5, 8.C.5, 8.D.5 What is a limit? What is a derivative? What role do derivatives and limits

More information

MATH 2300 review problems for Exam 3 ANSWERS

MATH 2300 review problems for Exam 3 ANSWERS MATH 300 review problems for Exam 3 ANSWERS. Check whether the following series converge or diverge. In each case, justify your answer by either computing the sum or by by showing which convergence test

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

AP Calculus AB. Practice Exam. Advanced Placement Program

AP Calculus AB. Practice Exam. Advanced Placement Program Advanced Placement Program AP Calculus AB Practice Exam The questions contained in this AP Calculus AB Practice Exam are written to the content specifications of AP Exams for this subject. Taking this

More information

1. [2.3] Techniques for Computing Limits Limits of Polynomials/Rational Functions/Continuous Functions. Indeterminate Form-Eliminate the Common Factor

1. [2.3] Techniques for Computing Limits Limits of Polynomials/Rational Functions/Continuous Functions. Indeterminate Form-Eliminate the Common Factor Review for the BST MTHSC 8 Name : [] Techniques for Computing Limits Limits of Polynomials/Rational Functions/Continuous Functions Evaluate cos 6 Indeterminate Form-Eliminate the Common Factor Find the

More information

Geometry Notes PERIMETER AND AREA

Geometry Notes PERIMETER AND AREA Perimeter and Area Page 1 of 57 PERIMETER AND AREA Objectives: After completing this section, you should be able to do the following: Calculate the area of given geometric figures. Calculate the perimeter

More information

Lesson A - Natural Exponential Function and Natural Logarithm Functions

Lesson A - Natural Exponential Function and Natural Logarithm Functions A- Lesson A - Natural Exponential Function and Natural Logarithm Functions Natural Exponential Function In Lesson 2, we explored the world of logarithms in base 0. The natural logarithm has a base of e.

More information

Section 3.7. Rolle s Theorem and the Mean Value Theorem. Difference Equations to Differential Equations

Section 3.7. Rolle s Theorem and the Mean Value Theorem. Difference Equations to Differential Equations Difference Equations to Differential Equations Section.7 Rolle s Theorem and the Mean Value Theorem The two theorems which are at the heart of this section draw connections between the instantaneous rate

More information

Student Performance Q&A:

Student Performance Q&A: Student Performance Q&A: 2008 AP Calculus AB and Calculus BC Free-Response Questions The following comments on the 2008 free-response questions for AP Calculus AB and Calculus BC were written by the Chief

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

MTH4100 Calculus I. Lecture notes for Week 8. Thomas Calculus, Sections 4.1 to 4.4. Rainer Klages

MTH4100 Calculus I. Lecture notes for Week 8. Thomas Calculus, Sections 4.1 to 4.4. Rainer Klages MTH4100 Calculus I Lecture notes for Week 8 Thomas Calculus, Sections 4.1 to 4.4 Rainer Klages School of Mathematical Sciences Queen Mary University of London Autumn 2009 Theorem 1 (First Derivative Theorem

More information

Worksheet for Week 1: Circles and lines

Worksheet for Week 1: Circles and lines Worksheet Math 124 Week 1 Worksheet for Week 1: Circles and lines This worksheet is a review of circles and lines, and will give you some practice with algebra and with graphing. Also, this worksheet introduces

More information

www.mathsbox.org.uk ab = c a If the coefficients a,b and c are real then either α and β are real or α and β are complex conjugates

www.mathsbox.org.uk ab = c a If the coefficients a,b and c are real then either α and β are real or α and β are complex conjugates Further Pure Summary Notes. Roots of Quadratic Equations For a quadratic equation ax + bx + c = 0 with roots α and β Sum of the roots Product of roots a + b = b a ab = c a If the coefficients a,b and c

More information

y cos 3 x dx y cos 2 x cos x dx y 1 sin 2 x cos x dx

y cos 3 x dx y cos 2 x cos x dx y 1 sin 2 x cos x dx Trigonometric Integrals In this section we use trigonometric identities to integrate certain combinations of trigonometric functions. We start with powers of sine and cosine. EXAMPLE Evaluate cos 3 x dx.

More information

6.6 The Inverse Trigonometric Functions. Outline

6.6 The Inverse Trigonometric Functions. Outline 6.6 The Inverse Trigonometric Functions Tom Lewis Fall Semester 2015 Outline The inverse sine function The inverse cosine function The inverse tangent function The other inverse trig functions Miscellaneous

More information

Lecture 5 : Continuous Functions Definition 1 We say the function f is continuous at a number a if

Lecture 5 : Continuous Functions Definition 1 We say the function f is continuous at a number a if Lecture 5 : Continuous Functions Definition We say the function f is continuous at a number a if f(x) = f(a). (i.e. we can make the value of f(x) as close as we like to f(a) by taking x sufficiently close

More information

Calculating Areas Section 6.1

Calculating Areas Section 6.1 A B I L E N E C H R I S T I A N U N I V E R S I T Y Department of Mathematics Calculating Areas Section 6.1 Dr. John Ehrke Department of Mathematics Fall 2012 Measuring Area By Slicing We first defined

More information

Area Between Curves. The idea: the area between curves y = f(x) and y = g(x) (if the graph of f(x) is above that of g(x)) for a x b is given by

Area Between Curves. The idea: the area between curves y = f(x) and y = g(x) (if the graph of f(x) is above that of g(x)) for a x b is given by MATH 42, Fall 29 Examples from Section, Tue, 27 Oct 29 1 The First Hour Area Between Curves. The idea: the area between curves y = f(x) and y = g(x) (if the graph of f(x) is above that of g(x)) for a x

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

a b c d e You have two hours to do this exam. Please write your name on this page, and at the top of page three. GOOD LUCK! 3. a b c d e 12.

a b c d e You have two hours to do this exam. Please write your name on this page, and at the top of page three. GOOD LUCK! 3. a b c d e 12. MA123 Elem. Calculus Fall 2015 Exam 3 2015-11-19 Name: Sec.: Do not remove this answer page you will turn in the entire exam. No books or notes may be used. You may use an ACT-approved calculator during

More information

Calculus with Analytic Geometry I Exam 10 Take Home part

Calculus with Analytic Geometry I Exam 10 Take Home part Calculus with Analytic Geometry I Exam 10 Take Home part Textbook, Section 47, Exercises #22, 30, 32, 38, 48, 56, 70, 76 1 # 22) Find, correct to two decimal places, the coordinates of the point on the

More information

Series FOURIER SERIES. Graham S McDonald. A self-contained Tutorial Module for learning the technique of Fourier series analysis

Series FOURIER SERIES. Graham S McDonald. A self-contained Tutorial Module for learning the technique of Fourier series analysis Series FOURIER SERIES Graham S McDonald A self-contained Tutorial Module for learning the technique of Fourier series analysis Table of contents Begin Tutorial c 004 g.s.mcdonald@salford.ac.uk 1. Theory.

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

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

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

4.4 Concavity and Curve Sketching

4.4 Concavity and Curve Sketching Concavity and Curve Sketching Section Notes Page We can use the second derivative to tell us if a graph is concave up or concave down To see if something is concave down or concave up we need to look at

More information

3.5: Issues in Curve Sketching

3.5: Issues in Curve Sketching 3.5: Issues in Curve Sketching Mathematics 3 Lecture 20 Dartmouth College February 17, 2010 Typeset by FoilTEX Example 1 Which of the following are the graphs of a function, its derivative and its second

More information

a. all of the above b. none of the above c. B, C, D, and F d. C, D, F e. C only f. C and F

a. all of the above b. none of the above c. B, C, D, and F d. C, D, F e. C only f. C and F FINAL REVIEW WORKSHEET COLLEGE ALGEBRA Chapter 1. 1. Given the following equations, which are functions? (A) y 2 = 1 x 2 (B) y = 9 (C) y = x 3 5x (D) 5x + 2y = 10 (E) y = ± 1 2x (F) y = 3 x + 5 a. all

More information

Able Enrichment Centre - Prep Level Curriculum

Able Enrichment Centre - Prep Level Curriculum Able Enrichment Centre - Prep Level Curriculum Unit 1: Number Systems Number Line Converting expanded form into standard form or vice versa. Define: Prime Number, Natural Number, Integer, Rational Number,

More information

Chapter (AB/BC, non-calculator) (a) Write an equation of the line tangent to the graph of f at x 2.

Chapter (AB/BC, non-calculator) (a) Write an equation of the line tangent to the graph of f at x 2. Chapter 1. (AB/BC, non-calculator) Let f( x) x 3 4. (a) Write an equation of the line tangent to the graph of f at x. (b) Find the values of x for which the graph of f has a horizontal tangent. (c) Find

More information

x), etc. In general, we have

x), etc. In general, we have BASIC CALCULUS REFRESHER. Introduction. Ismor Fischer, Ph.D. Dept. of Statistics UW-Madison This is a very condensed and simplified version of basic calculus, which is a prerequisite for many courses in

More information

Calculus with Parametric Curves

Calculus with Parametric Curves Calculus with Parametric Curves Suppose f and g are differentiable functions and we want to find the tangent line at a point on the parametric curve x f(t), y g(t) where y is also a differentiable function

More information

APPLICATION OF DERIVATIVES

APPLICATION OF DERIVATIVES 6. Overview 6.. Rate of change of quantities For the function y f (x), d (f (x)) represents the rate of change of y with respect to x. dx Thus if s represents the distance and t the time, then ds represents

More information

Section 4.4. Using the Fundamental Theorem. Difference Equations to Differential Equations

Section 4.4. Using the Fundamental Theorem. Difference Equations to Differential Equations Difference Equations to Differential Equations Section 4.4 Using the Fundamental Theorem As we saw in Section 4.3, using the Fundamental Theorem of Integral Calculus reduces the problem of evaluating a

More information

Applications of Integration to Geometry

Applications of Integration to Geometry Applications of Integration to Geometry Volumes of Revolution We can create a solid having circular cross-sections by revolving regions in the plane along a line, giving a solid of revolution. Note that

More information

CALCULUS 2. 0 Repetition. tutorials 2015/ Find limits of the following sequences or prove that they are divergent.

CALCULUS 2. 0 Repetition. tutorials 2015/ Find limits of the following sequences or prove that they are divergent. CALCULUS tutorials 5/6 Repetition. Find limits of the following sequences or prove that they are divergent. a n = n( ) n, a n = n 3 7 n 5 n +, a n = ( n n 4n + 7 ), a n = n3 5n + 3 4n 7 3n, 3 ( ) 3n 6n

More information

100. In general, we can define this as if b x = a then x = log b

100. In general, we can define this as if b x = a then x = log b Exponents and Logarithms Review 1. Solving exponential equations: Solve : a)8 x = 4! x! 3 b)3 x+1 + 9 x = 18 c)3x 3 = 1 3. Recall: Terminology of Logarithms If 10 x = 100 then of course, x =. However,

More information

Mathematics Placement Examination (MPE)

Mathematics Placement Examination (MPE) Practice Problems for Mathematics Placement Eamination (MPE) Revised August, 04 When you come to New Meico State University, you may be asked to take the Mathematics Placement Eamination (MPE) Your inital

More information

Function Name Algebra. Parent Function. Characteristics. Harold s Parent Functions Cheat Sheet 28 December 2015

Function Name Algebra. Parent Function. Characteristics. Harold s Parent Functions Cheat Sheet 28 December 2015 Harold s s Cheat Sheet 8 December 05 Algebra Constant Linear Identity f(x) c f(x) x Range: [c, c] Undefined (asymptote) Restrictions: c is a real number Ay + B 0 g(x) x Restrictions: m 0 General Fms: Ax

More information

TRIGONOMETRIC IDENTITIES

TRIGONOMETRIC IDENTITIES Integration TRIGONOMETRIC IDENTITIES Graham S McDonald and Silvia C Dalla A self-contained Tutorial Module for practising integration of expressions involving products of trigonometric functions such as

More information

Derivatives and Graphs. Review of basic rules: We have already discussed the Power Rule.

Derivatives and Graphs. Review of basic rules: We have already discussed the Power Rule. Derivatives and Graphs Review of basic rules: We have already discussed the Power Rule. Product Rule: If y = f (x)g(x) dy dx = Proof by first principles: Quotient Rule: If y = f (x) g(x) dy dx = Proof,

More information

SECTION 0.11: SOLVING EQUATIONS. LEARNING OBJECTIVES Know how to solve linear, quadratic, rational, radical, and absolute value equations.

SECTION 0.11: SOLVING EQUATIONS. LEARNING OBJECTIVES Know how to solve linear, quadratic, rational, radical, and absolute value equations. (Section 0.11: Solving Equations) 0.11.1 SECTION 0.11: SOLVING EQUATIONS LEARNING OBJECTIVES Know how to solve linear, quadratic, rational, radical, and absolute value equations. PART A: DISCUSSION Much

More information

A couple of Max Min Examples. 3.6 # 2) Find the maximum possible area of a rectangle of perimeter 200m.

A couple of Max Min Examples. 3.6 # 2) Find the maximum possible area of a rectangle of perimeter 200m. A couple of Max Min Examples 3.6 # 2) Find the maximum possible area of a rectangle of perimeter 200m. Solution : As is the case with all of these problems, we need to find a function to minimize or maximize

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

Using a table of derivatives

Using a table of derivatives Using a table of derivatives In this unit we construct a Table of Derivatives of commonly occurring functions. This is done using the knowledge gained in previous units on differentiation from first principles.

More information

Name Calculus AP Chapter 7 Outline M. C.

Name Calculus AP Chapter 7 Outline M. C. Name Calculus AP Chapter 7 Outline M. C. A. AREA UNDER A CURVE: a. If y = f (x) is continuous and non-negative on [a, b], then the area under the curve of f from a to b is: A = f (x) dx b. If y = f (x)

More information