Final Exam Study Guide

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1 Final Exam Study Guide Final Exam Coverage: Sections , , 10.7, , , , , and 14.1 Sections/topics NOT on the exam: Sections 10.3 (Continuity, it definition of the derivative from 10.4, 10.6 (Differentials, 11.1 (The constant e and Continuous Compound Interest, 11.5 (Implicit Differentiation, 11.6 (Related Rates, 11.7 (Elasticity of Demand, Riemann sums from 13.4, 14.2 (Applications in Business and Economics, 15.1 (Functions of Several Variables, and 15.2 (Partial Derivatives Section 10.1: Introduction to Limits List of Topics for the Final Exam ˆ Know what a it means (you don t need to know the exact definition but you should have a general idea of what is meant by taking a it of a function ˆ One-sided its ˆ When does a it x c f(x fail to exist? ˆ Know how to use a graph of a function to determine various its or show that a it does not exist ˆ Know properties of its and how to use them to find its ˆ To find a it as x c of a rational expression p(x/q(x, when are you allowed to just plug in c into the expression to get the it? ˆ Know what is meant by a 0 0 indeterminate form and what this means about a it ˆ Know how to use factoring to find a it ˆ Know when a it of a rational expression p(x/q(x does not exist (or, as discussed in later sections, is + or. Section 10.2: Infinite Limits and Limits at Infinity ˆ Infinite its (such as x c f(x = or x c f(x = ˆ Know how to identify all vertical asymptotes of a function ˆ Limits at infinity, such as f(x = L or f(x = L. We could also replace L in these x x expressions with or. ˆ Know how to find horizontal asymptotes of a function ˆ Know how to take its at infinity of polynomial functions and rational functions. Section 10.4: The Derivative ˆ You will not need to know the it definition of the derivative, or how to use it, for the final exam. ˆ You should know the different interpretations of the derivative. ˆ Know how to find the equation of the tangent line to a function at a point. Section 10.5: Basic Differentiation Properties ˆ power rule ˆ constant multiple property ˆ sum and difference properties Section 10.7: Marginal Analysis in Business and Economics 1

2 ˆ Cost, revenue, profit functions - know what these are and how to use them in applications ˆ Marginal cost, marginal revenue, marginal profit - know how to find these and how to interpret them ˆ Price-demand equation ˆ Know how to use the price-demand equation to find revenue ˆ Know the relationship between cost, revenue, and profit ˆ Know how to indicate break-even points on a graph and what these points mean ˆ Know how to use a graph of revenue and cost to find regions of profit and loss Section 11.2: Derivatives of Exponential and Logarithmic Functions ˆ Know derivative of e x and ln(x Section 11.3: Derivatives of Products and Quotients ˆ Product rule ˆ Quotient rule Section 11.4: Chain Rule ˆ Know how and when to use the chain rule Section 12.1: First Derivative and Graphs ˆ partition numbers - know how to find them and how they are useful ˆ Know how to make a sign chart for f (x ˆ Know how to use the sign chart to find local maxes/mins ˆ Know how to find the intervals on which a function is increasing/decreasing Section 12.2: Second Derivative and Graphs ˆ Know how to make a sign chart for f (x ˆ Know how to use the sign chart to find inflection points and to find on what intervals a function is concave up/down ˆ Know how to use information about concavity and increasing/decreasing to help sketch a graph Section 12.3: L Hopital s Rule ˆ Know what is meant by 0 0 and indeterminate forms ˆ Know how to determine if L Hopital s Rule can be applied to find a it of a function ˆ Know how to use L Hopital s rule to find the it of a function (assuming L Hopital s rule can be used Section 12.4: Curve-Sketching Techniques ˆ Know how to apply the graphing procedure taught in this section which includes Looking at f(x and finding its domain, intercepts, asymptotes (horizontal and vertical, its as x approaches an asymptote Analyzing the first derivative f (x (finding where f(x is increasing/decreasing, critical points, the local mins/maxes Analyzing the second derivative f (x (finding the inflection points, where f(x is concave up/down Sketching the graph using all of the info obtained 2

3 ˆ Be able to draw a sketch of a graph that satisfies certain listed info, which may include info about asymptotes, its, intervals on which the function is increasing/decreasing, and intervals on which the function is concave up/down. Section 12.5: Absolute Maxima and Minima ˆ Know what is meant by an absolute maximum, an absolute minimum, and absolute extrema ˆ Know how to find (and show that you ve found an absolute extrema, either on a closed interval or an open one Section 12.6: Optimization ˆ Given an optimization problem, know how to Draw a picture (if necessary and introduce your variables Find your constraint equation (an equation that tells how the variables you introduced previously are related Write down your objective function (the function you are trying to maximize or minimize as a function of one variable only Maximize (or minimize the objective function, and show that you ve done so by including a sign chart. Finish answering the question Section 13.1: Antiderivatives and Indefinite Integrals ˆ Antiderivatives (know what they are and how to find them ˆ Know how to use integral notation f(x dx to represent the family of antiderivatives of f(x ˆ Properties of indefinite integrals Section 13.2: Integration by Substitution ˆ Know how to use integration by substitution (also known as u-substitution to find an integral Section 13.4: The Definite Integral ˆ You don t need to know how to compute Riemann sums ˆ You do need to know what the definite integral represents geometrically in terms of areas. ˆ Given areas of certain shaded areas of a graph, know how to use these to find a definite integral ˆ Properties of definite integrals Section 13.5: The Fundamental Theorem of Calculus ˆ Know what the Fundamental Theorem of Calculus says and know how to apply it to find a definite integral Section 14.1: Areas Between Curves ˆ If f(x g(x over an interval [a, b], know how to find the area bounded by these two functions over [a, b]. 3

4 Practice Problems Note: These problems are just some of the types of problems that might appear on the exam. However, to fully prepare for the exam, in addition to making sure you can do all of these problems, you should review your homework and class notes as well. Section 10.1: Introduction to Limits 1. The graph of f is shown below. Use the graph of f to evaluate the indicated its and function values. If a it or function value does not exist, write DNE. (a f(x x 2 + (b x 2 f(x (c f(x x 2 (d f( 2 (e x 0 f(x (f f(2 (g (h f(x x 2 x 2 + f(x (i x 2 f(x 2. Suppose x 1 f(x = 3 and x 1 g(x = 2. Find (a x 1 3f(x ( (b g(x f(x x 1 (c x 1 3 f(x 4g(x 2 3. Find each it ( (a x 2 3 x 3 (b 5 x 1 (c x x 2 (d x x 1 x 4. For each it, is the it a 0 0 not exist. (a x 7 (x 7 2 x 2 4x 21 (b x 2 x 2 x + 2 (c x 4 3x + 12 x 2 4 indeterminant form? Find the it or explain why it does 4

5 (d x 3 x 2 9 x 3 5. Circle or to the following statements. f(x (a If f(x = 0 and g(x = 0, then x 1 x 1 x 1 g(x does not exist. f(x (b If f(x = 2 and g(x = 2, then x 1 x 1 x 2 g(x = 1 (c In order for f(x to exist and be equal to 2, it must be the case that f(x = 2 x 1 x 1 and f(x = x 1 + 2ṫrue (d If f is a function such that f(2 exists, then x 2 f(x exists. (e If f is a polynomial, then x c f(x = f(c for every real number c. Section 10.2: Infinite Limits and Limits at Infinity 1. Suppose f(x = 1. Find each it. Write,, or DNE where appropriate. x 5 (a f(x x 5 (b f(x x 5 + (c f(x x 5 2. Suppose f(x = 3 x. Find each it. Write,, or DNE where appropriate. x + 2 (a (b (c f(x x 2 x 2 + f(x f(x x 2 3. Identify all vertical and horizontal asymptotes of the following functions. (a f(x = 1 x + 3 (b f(x = x2 2x 3 x Find the following its. Write or where appropriate. ( (a x 2 2x 1 x ( (b x 3 x x ( (c x 3 + 2x + x x 5

6 5. Circle or to the following statements. (a A polynomial function of degree greater than or equal to 1 has neither horizontal nor vertical asymptotes. (b A rational function always has at least one vertical asymptote. (c A rational function has at most one horizontal asymptote. 6. A theorem states that for n 1 and a n 0 (a n x n + a n 1 x n a 0 = ± x What conditions must n and a n satisfy for the it to be +? Section 10.4: The Derivative 1. Write down two interpretations of f (a, the derivative of a function f(x evaluated at x = a. 2. Explain what f (2 = 3 tells us about a function f(x. 3. Find the equation of the tangent line to f(x = 2 x 2 at x = 1. Section 10.5: Basic Differentiation Properties 1. If C is any constant, what is d dx (C? 2. Write down the power rule. 3. If f (x = 2x and g (x = x 2, what is d (2f(x 1 dx 4 g(x? 4. Find the derivative of the following functions. Do not simplify. (a f(x = 7 (b f(x = 2x + 3 (c f(x = 2x 2 x + 1 (d y = x + x 3/4 (e y = 1 5x 2x 3 3 x 2/3 Section 10.7: Marginal Analysis in Business and Economics 1. What is the relationship between the cost function C(x, the revenue function R(x, and the profit function P (x? 2. Suppose the total cost of producing x items is C(x = x 1 2 x2 and the total profit from the sale of x items is P (x = 40x 1 6 x2 100 (a Find the marginal cost function. (b Find the marginal cost when 10 items are produced and interpret the result. (c Find the marginal profit function. (d Find the marginal profit when 10 items are produced and interpret. (e Find the revenue function. 6

7 (f Find the revenue gained from selling 10 items. 3. A company is producing couches and their total cost function is C(x = 100, x. Their price-demand equation is x = p. (a Express the price p as a function of the demand x, then find the domain of this function. (b Find the marginal cost. (c Find the revenue function as a function of x and state its domain. (d Find the marginal revenue. (e Find R (2000 and R (5000 and interpret. (f Graph the cost function and the revenue function. Indicate the break-even points on the graph. (g The break-even points are about (764, and (5236, Use these to write down the intervals of loss and the interval of profit. (h Find the profit function P (x. Section 11.2: Derivatives of Exponential and Logarithmic Functions 1. Find d ( 5e x ln(x + 2. dx 2. Use log properties to find derivatives of the following functions. (Note: If you use the log properties you won t need to use the chain rule. (a f(x = x 4 ln(x 5 (b y = ln(xe x ( x (c y = ln 5 x Section 11.2: Derivatives of Exponential and Logarithmic Functions 1. Find d ( 5e x ln(x + 2. dx 2. Use log properties to find derivatives of the following functions. (Note: If you use the log properties you won t need to use the chain rule. (a f(x = x 4 ln(x 5 (b y = ln(xe x ( x (c y = ln 5 x Section 11.3: Derivatives of Products and Quotients 1. Fill in the blanks: (a The product rule says that given two differentiable functions F (x and S(x, the derivative of their product is d ( F (xs(x = dx (b The quotient rule says that given two differentiable functions T (x and B(x, the derivative of their quotient is ( d T (x = dx B(x 2. Find d (x 2 e x. Do not simplify. dx 3. If f(x = x2 4x 3, find f (x. Do not simplify. 4. Find f (z if f(z = z ln(z. Do not simplify. z 3 7

8 5. Find d dt ( e t t 2 3 ln(t + t 3/2. Do not simplify. 6. Find f (x if f(x = ex ln(x 1 e x. Do not simplify. Section 11.4: The Chain Rule 1. Find the indicated derivatives. You don t have to simplify. (a f (x if f(x = (3x 2 2x 10 (b dy dx if y = ln(x2 + 1 (c y if y = e x4 (d f (x if f(x = (x 2 ln x 5 (e f (w if f(w = (f d ( (1 t 2 e t3 t dt 1 3 (2w ew 4 Section 12.1: First Derivative and Graphs 1. For each function, find the intervals on which f(x is increasing and the intervals on which f(x is decreasing. Find any local maxes or mins. (a f(x = 2x 2 8x + 9 (b f(x = x x 2 (c f(x = x 3 + 3x 2 + 3x 2. Use the given info to sketch the graph of f. Assume its domain is (,. ˆ f( 2 = 4, f(0 = 0, f(2 = 4 ˆ f ( 2 = 0, f (0 = 0, f (2 = 0 ˆ f (x > 0 on (, 2 and (2, ˆ f (x < 0 on ( 2, 0 and (0, 2 Section 12:2: Second Derivative and Graphs 1. Find the inflection points of f(x = ln(x 2 4x Suppose f(x = x 1/3. Find the inflection point(s of f(x. Also, find the intervals on which f(x is concave up and the intervals on which f(x is concave down. 3. Suppose f(x = x 4 2x 3. Find (a The domain of f(x. (b The intercepts of f(x. (c Make a sign chart for f (x. Find its critcal numbers. Find any local extrema. List the intervals on which f is increasing and the intervals on which f is decreasing. (d Make a sign chart for f (x. Find any inflection point(s and list the intervals on which f(x is concave up or concave down. (e Sketch the graph of f. Section 12.3: L Hopital s Rule 1. For each it, determine if the it is a 0 0 indeterminate form, a indeterminate form, or neither. Can L Hopital s Rule be applied to find the it? If not, explain why not. If it can be applied, use L Hopital s Rule to find the it. (a x 0 ln(1 + x 2 x 4 ln x (b x 1 x 8

9 (c ln(1 + 2e x x e x e x (d x ln x (e x 0 3x + 1 e 3x x 2 2. For each it listed, first (i find the it using previous methods, and then (ii use L Hopital s Rule to find the it. 2x 2 1 (a x x x 1 (b x x 2 4 (c x 3 x 2 9 x + 3 Section 12.4: Curve Sketching Techniques 1. Use the graphing strategy taught in this section to analyze and graph the following functions. (a g(x = 4x + 3 x 2 (b f(x = xe 0.5x (c f(x = x ln x 2. Sketch a function which satisfies the following properties. ˆ f( 3 = 1, f(0 = 0, f(3 = 1 ˆ f (x < 0 on (, 2 and (2, ˆ f (x > 0 on ( 2, 2 ˆ f (x < 0 on (, 2 and ( 2, 0 ˆ f (x > 0 on (0, 2 and (2, ˆ vertical asymptotes: x = 2, x = 2 ˆ horizontal asymptotes: y = 0 Section 12.5: Absolute Maxima and Minima 1. Find the absolute maximum and absolute minimum of f(x = x 3 12x on each of the following intervals: (a [ 3, 3] (b [ 3, 1] 2. Find the absolute extrema of each function on (0,. (a f(x = 12 x 5 x (b f(x = f ln(x x Section 12.6: Optimization 1. Find the greatest possible product of two numbers given that the sum of the two numbers equals Find two numbers whose difference is 80 and whose product is a minimum. 3. Find the area of the largest rectangle that can be made with a perimeter of 56 ft. 4. Find the largest possible perimeter of a rectangle whose area is 400 ft Suppose you want to fence a rectangular area with one side against a barn (so no fencing needs to be used for that side. If the amount of fencing to be used is 40 ft, find the dimensions of the rectangle that has the maximum area. 9

10 Section 13.1: Antiderivatives and Indefinite Integrals 1. What does it mean for F (x to be an antiderivative of f(x? 2. If n is any real number, what do the family of antiderivatives of x n look like? 3. Find an antiderivative of f(x = 3x Find an antiderivative of f(x = 2x Find each indefinite integral. ( (a x 2 5 x + 4ex dx ( x 2 (b dx x 2 ( 1 (c x x 5 dx (d x(x dx 6. Circle or to the following statements. (a Every function has an infinite number of antiderivatives. (b An antiderivative of f(x = 1 x is F (x = ln x (c An antiderivative of f(x = 2x is x 2 + x. (d The constant function f(x = 0 is an antiderivative of itself. (e The function f(x = 5e x is an antiderivative of itself. (f If n is an integer, then Section 13.2: Integration by Substitution 1 n + 1 xn+1 is an antiderivative of x n. 1. Use integration by substitution to find each integral. (a (b (c 2 x dx x x dx x 2 x3 + 5 dx 2. Integrate. (a e 3x dx x (b x 2 9 dx (c 5t 2 (t dt 10

11 Matched Problem 5 on p. 731, answer on p. 736 Section 13.4: The Definite Integral 1. Find the area under the graph of f(x = x but above the x-axis between x = 1 and x = Consider the graph of f(x shown below, where A, B, and C represent the shaded regions shown. Suppose area of A = 1.5 area of B = 2.5 area of C = 0.5 Find the following definite integrals. (a (b (c (d (e f(x dx f(x dx f(x dx f(x dx f(x dx 3. Find the following definite integrals. (a (b (c (d (e ( x 3 x dx ( 6x 2 1 dx x x x 2 2 dx 4 x dx 2x x2 1 dx 11

12 Section 14.1: Area Between Curves 1. Find the area bounded by the graphs of f(x = x and g(x = x 2 + 4x as shown below. 2. Find the area between the graph of f(x = x 2 x 2 and the x-axis over [ 2, 2]. The graph of f(x has been graphed below. The area will be the sum of the areas of the two shaded regions A and B as shown. 12