Midterm Study Guide and Practice Problems

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1 Midterm Study Guide and Practice Problems Coverage of the midterm: Sections , Sections or topics NOT on the midterm: Section 11.1 (The constant e and continuous compound interest, Section 11.7 (Elasticity of Demand Section 10.1: Introduction to Limits List of Topics for the Final Exam ˆ Know what a limit means (you don t need to know the exact definition but you should have a general idea of what is meant by taking a limit of a function ˆ One-sided limits ˆ When does a limit lim x c f(x fail to exist? ˆ Know how to use a graph of a function to determine various limits or show that a limit does not exist ˆ Know properties of limits and how to use them to find limits ˆ To find a limit 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 limit? ˆ Know what is meant by a 0 0 indeterminate form and what this means about a limit ˆ Know how to use factoring to find a limit ˆ Know when a limit 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 limits (such as lim x c f(x = or lim x c f(x = ˆ Know how to identify all vertical asymptotes of a function ˆ Limits at infinity, such as lim f(x = L or lim 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 limits at infinity of polynomial functions and rational functions. Section 10.3: Continuity ˆ Know the three properties that must be satisfied for a function to be continuous at a point ˆ Know how to find where a function is not continuous from looking at its graph ˆ Know how to find where a polynomial or a rational function is continuous ˆ Know how to find where a function in the form n f(x is continuous ˆ Know how to solve an inequality of the form f(x 0, f(x > 0, f(x < 0, or f(x 0 where f(x is a rational function Section 10.4: The Derivative ˆ You don t need to memorize the definition of the derivative as a limit. If a question asks you to use the definition of the derivative to find a derivative, the formula will be provided. However, you do need to know how to use it. ˆ The different interpretations of the derivative ˆ Know how to find the equation of the tangent line to a function at a point. 1

2 Section 10.5: Basic Differentiation Properties ˆ power rule ˆ constant multiple property ˆ sum and difference properties Section 10.6: Differentials ˆ Know the formula for dy in terms of x and ˆ Know how to use dy to approximate the actual change y when x changes from some number x 1 to some number x 2. Section 10.7: Marginal Analysis in Business and Economics 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 ˆ You won t need to know how to find derivatives of things like b x and log b (x where b e. 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 11.5: Implicit Differentiation ˆ Know how to use implicit differentiation to find dy defined in an equation where y is implicitly ˆ Know how to evaluate your answer for dy that point at a given point to find the slope of a curve at ˆ Know how to find the tangent line to a curve at a point Section 11.6: Related Rates ˆ Given some related rates problem, know how to Sketch a figure for the problem Set up an equation that relates all of the variables Use implicit differentiation to differentiate both sides of the equation (normally with respect to time t Know how to solve for the appropriate rate and evaluate it by plugging in the appropriate numbers for the other variables/rates 2

3 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. Note problems different than the ones listed here may appear on your exam. However, at least one of the problems listed here will appear as a problem on the midterm. Section 10.1: Introduction to Limits 1. The graph of f is shown below. Use the graph of f to evaluate the indicated limits and function values. If a limit or function value does not exist, write DNE. (a lim f(x x 2 + (b lim x 2 f(x (c lim f(x x 2 (d f( 2 (e lim x 0 f(x (f f(2 (g (h lim f(x x 2 lim x 2 + f(x (i lim x 2 f(x 2. Suppose lim f(x = 3 and lim g(x = 2. Find (a lim 3f(x ( (b lim g(x f(x (c lim 3 f(x 4g(x 2 3. Find each limit ( (a lim x 2 3 x 3 (b lim 5 (c lim x x 2 (d x lim x 1 x 4. For each limit, is the limit a 0 0 not exist. (x 7 2 (a lim x 7 x 2 4x 21 x 2 (b lim x 2 x + 2 indeterminant form? Find the limit or explain why it does 3

4 3x + 12 (c lim x 4 x 2 4 x 2 9 (d lim x 3 x 3 5. Circle or to the following statements. f(x (a If lim f(x = 0 and lim g(x = 0, then lim g(x does not exist. f(x (b If lim f(x = 2 and lim g(x = 2, then lim x 2 g(x = 1 (c In order for lim f(x to exist and be equal to 2, it must be the case that lim f(x = 2 and lim f(x = + 2ṫrue (d If f is a function such that f(2 exists, then lim x 2 f(x exists. (e If f is a polynomial, then lim 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 limit. Write,, or DNE where appropriate. x 5 (a lim f(x x 5 (b lim f(x x 5 + (c lim f(x x 5 2. Suppose f(x = 3 x. Find each limit. Write,, or DNE where appropriate. x + 2 (a (b (c lim f(x x 2 lim f(x x 2 + lim 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 limits. Write or where appropriate. ( (a lim x 2 2x 1 x ( (b lim x 3 x x ( (c x 3 + 2x + x lim x 4

5 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 = ± lim x What conditions must n and a n satisfy for the limit to be +? Section 10.3: Continuity 1. Write down the three properties that must be satisfied in order for a function f(x to be continuous at the point x = a. 2. Suppose f(x is the function whose graph is shown below. Where is f(x not continuous? Explain why it is not continuous at each of these points by explaining, for each point, why at least one of the properties which you ve listed in the previous problem fails to be satisfied. 3. Where are the following functions continuous? Use interval notation to write your answer. (a f(x = x 2 x + 2 (b f(x = 1 x 5 (c f(x = x 4 x 2 16 (d f(x = x x (e f(x = x 5 (f f(x = 9 x Solve the following inequalities. Find partition numbers and make a sign chart as part of your explanation for each. 5

6 (a x 2 x + 4 > 0 (b x 2 4 x 2 + 5x 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. Recall that the definition of the derivative of a function f(x is f (x = lim h 0 f(x + h f(x h (a Use the definition of the derivative to prove that the derivative of f(x = 2 x 2 is f (x = 2x. (b 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 (C? 2. Write down the power rule. 3. If f (x = 2x and g (x = x 2, what is d (2f(x 1 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 10.6: Differentials 1. Let f(x = 1 2 x2 + x. (a Find dy as a function of x and. (b If x changes from 1 to 1.1, use part (a to approximate y, the corresponding change in the y-values. Do not find y exactly. 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. (f Find the revenue gained from selling 10 items. 6

7 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 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 = (b The quotient rule says that given two differentiable functions T (x and B(x, the derivative of their quotient is ( d T (x = B(x 2. Find d (x 2 e x. Do not simplify. 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. 5. Find d dt ( e t t 2 3 ln(t + t 3/2 z 3. 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 if y = ln(x

8 (c f 1 (w if f(w = 3 (2w ew 4 (d d ( (1 t 2 e t3 t dt Section 11.5: Implicit Differentiation 1. Use implicit differentiation to find dy if 2y xy2 + xy = Suppose e 2x y x 5 + x ln y = 0. Find dy. 3. A curve is described by the equation y + x xy = 1. (a Use implicit differentiation to find dy. (b Find dy. That is, find dy when x = 1 and y = 1. (1,1 (c Find the equation of the tangent line to the curve at (1, 1. Section 11.6: Related Rates 1. A 10-ft ladder is placed against a vertical wall. Suppose that the bottom of the ladder slides away from the wall at a constant rate of 3 ft/s. How fast is the top of the ladder sliding down the wall when the bottom is 6 feet from the wall? 2. The area of a square is decreasing at a rate of 2 ft 2 /min. When the area of the square is 9 ft 2, how fast is the side of the square decreasing? 3. A ball is shrinking at a constant rate of 2 m 3 /min. Find the rate at which the radius decreases when the radius is 1 m. Recall that the volume of a sphere is V = 4 3 πr3. 4. Suppose Jack and Jill both start at a point P and leave at the same time. Jack walks north at a rate of 2 mi/hr. At the same time, Judy walks east from P at a rate of 3 mi/hr. After 3 hours, how fast is the distance between Jack and Jill increasing? 5. Suppose one leg of a right triangle is increasing at a rate of 2 ft/min. The other leg remains at a length of 4 ft. How fast is the area of the triangle increasing? Recall that the area of a triangle is A = 1 (base (height Suppose that for a company manufacturing flash drives, the cost, revenue, and profit equations are given by C = x R = 10x 0.001x 2 P = R C where the production output in 1 week is x flash drives. If production is increasing at a rate of 500 flash drives per week when production is 6, 000 flash drives, find the rate of increase in (a Cost (b Revenue (c Profit 8