Social Security Assets The table gives actual and

Problem 2E a. Suppose . Use the graph of g(x) to find g?(0). ________________ b. Explain why the derivative of a function does not exist at a point where the tangent line is vertical.

Problem 5E Estimate the slope of the tangent line to the curve at the given point (x, y).

Problem 3E If , where is f not differentiable?

Problem 4E If the rate of change of f(x) is zero when x = a, what can be said about the tangent line to the graph of f(x) at x = a?

Problem 6E Estimate the slope of the tangent line to the curve at the given point (x, y).

Problem 7E Estimate the slope of the tangent line to the curve at the given point (x, y).

Problem 8E Estimate the slope of the tangent line to the curve at the given point (x, y).

Problem 9E Estimate the slope of the tangent line to the curve at the given point (x, y).

Problem 10E Estimate the slope of the tangent line to the curve at the given point (x, y).

Problem 11E Using the definition of the derivative, find f?(x). Then find f?(?2) f?(0), and f?(3) when the derivative exists. f(x) = 3x ? 7

Problem 12E Using the definition of the derivative, find f?(x). Then find f?(?2) f?(0), and f?(3) when the derivative exists. f(x) = ?2x + 5

Problem 13E Using the definition of the derivative, find f?(x). Then find f?(?2), f?(0), and f?(3) when the derivative exists. (Hint for Exercise: In Step 3, multiply numerator and denominator by f(x) = ?4x2 + 9x + 2

Problem 14E Using the definition of the derivative, find f?(x). Then find f?(?2), f?(0), and f?(3) when the derivative exists. (Hint for Exercise: In Step 3, multiply numerator and denominator by f(x) = 6x2 ? 5x ? 1

Problem 15E Using the definition of the derivative, find f?(x). Then find f?(?2), f?(0), and f?(3) when the derivative exists. (Hint for Exercise: In Step 3, multiply numerator and denominator by f(x) = 12/x

Problem 17E Using the definition of the derivative, find f?(x). Then find f?(?2), f?(0), and f?(3) when the derivative exists. (Hint for Exercise: In Step 3, multiply numerator and denominator by

Problem 16E Using the definition of the derivative, find f?(x). Then find f?(?2), f?(0), and f?(3) when the derivative exists. (Hint for Exercise: In Step 3, multiply numerator and denominator by f(x) = 3/x

Problem 18E Using the definition of the derivative, find f?(x). Then find f?(?2), f?(0), and f?(3) when the derivative exists. (Hint for Exercise: In Step 3, multiply numerator and denominator by

Problem 19E Using the definition of the derivative, find f?(x). Then find f?(?2), f?(0), and f?(3) when the derivative exists. (Hint for Exercise: In Step 3, multiply numerator and denominator by f(x) = 2x3 + 5

Problem 20E Using the definition of the derivative, find f?(x). Then find f?(?2), f?(0), and f?(3) when the derivative exists. (Hint for Exercise: In Step 3, multiply numerator and denominator by f(x) = 4x3 ? 3

Problem 21E For the function, find (a) the equation of the secant line through the points where x has the given values, and (b) the equation of the tangent line when xhas the first value. f(x) = x2 + 2x;x = 3, x = 5

Problem 22E For the function, find (a) the equation of the secant line through the points where x has the given values, and (b) the equation of the tangent line when xhas the first value. f(x) = 6 ? x2;x = ?1, x = 3

Problem 23E For the function, find (a) the equation of the secant line through the points where x has the given values, and (b) the equation of the tangent line when xhas the first value. f(x) = 5/x;x = 2, x = 5

Problem 24E For the function, find (a) the equation of the secant line through the points where x has the given values, and (b) the equation of the tangent line when xhas the first value. f(x) = ?3/(x + 1);x = 1, x = 5

Problem 26E For the function, find (a) the equation of the secant line through the points where x has the given values, and (b) the equation of the tangent line when xhas the first value.

Problem 25E For the function, find (a) the equation of the secant line through the points where x has the given values, and (b) the equation of the tangent line when xhas the first value.

Problem 27E Use a graphing calculator to find f?(2), f?(16), and f?(?3) for the following when the derivative exists. f(x) = ?4x2 + 11x

Problem 28E Use a graphing calculator to find f?(2), f?(16), and f?(?3) for the following when the derivative exists. f(x) = 6x2 ? 4x

Problem 29E Use a graphing calculator to find f?(2), f?(16), and f?(?3) for the following when the derivative exists. f(x) = ex

Problem 30E Use a graphing calculator to find f?(2), f?(16), and f?(?3) for the following when the derivative exists. f(x) = ln|x|

Problem 31E Use a graphing calculator to find f?(2), f?(16), and f?(?3) for the following when the derivative exists.

Problem 32E Use a graphing calculator to find f?(2), f?(16), and f?(?3) for the following when the derivative exists.

Problem 33E Use a graphing calculator to find f?(2), f?(16), and f?(?3) for the following when the derivative exists.

Problem 34E Use a graphing calculator to find f?(2), f?(16), and f?(?3) for the following when the derivative exists.

Problem 35E Find the x-values where the following do not have derivatives.

Problem 36E Find the x-values where the following do not have derivatives.

Problem 37E Find the x-values where the following do not have derivatives.

Problem 38E Find the x-values where the following do not have derivatives.

Problem 39E For the function shown in the sketch, give the intervals or points on the x-axis where the rate of change of f(x) with respect to x is a. positive; b. negative; c. zero.

Problem 40E In Exercise, tell which graph, a or b, represents velocity and which represents distance from a starting point. (Hint: Consider where the derivative is zero, positive, or negative.) a. b.

Problem 41E In Exercise, tell which graph, a or b, represents velocity and which represents distance from a starting point. (Hint: Consider where the derivative is zero, positive, or negative.) a. b.

Problem 42E In Exercise, find the derivative of the function at the given point. a. Approximate the definition of the derivative with small values of h. b.Use a graphing calculator to zoom in on the function until it appears to be a straight line, and then find the slope of that line. f(x) = xx;a = 2

Problem 43E In Exercise, find the derivative of the function at the given point. a. Approximate the definition of the derivative with small values of h. b.Use a graphing calculator to zoom in on the function until it appears to be a straight line, and then find the slope of that line. f(x) = xx;a = 3

Problem 44E In Exercise, find the derivative of the function at the given point. a. Approximate the definition of the derivative with small values of h. b.Use a graphing calculator to zoom in on the function until it appears to be a straight line, and then find the slope of that line. f(x) = x1/x;a = 2

Problem 45E In Exercise, find the derivative of the function at the given point. a. Approximate the definition of the derivative with small values of h. b.Use a graphing calculator to zoom in on the function until it appears to be a straight line, and then find the slope of that line. f(x) = x1/x; a = 3

Problem 46E For each function in Column A, graph [f(x + h) ? f(x)]/h for a small value of h on the window [ ? 2, 2] by [ ? 2, 8]. Then graph each function in Column B on the same window. Compare the first set of graphs with the second set to choose from Column B the derivative of each of the functions in Column A.

Problem 47E Explain why should give a reasonable approximation of f?(x) when f?(x) exists and h is small.

Problem 48E a. For the function f(x) = ?4x2 + 11x, find the value of f?(3), as well as the approximation using and using the formula in with h = 0.1. ________________ b. Repeat part a using h = 0.01. ________________ c. Repeat part a using the function f(x) = ?2/x and h = 0.1. ________________ d. Repeat part c using h = 0.01. ________________ e. Repeat part a using the function and h = 0.1. ________________ f. Repeat part e using h = 0.01. ________________ g. Using the results of parts a through f, discuss which approximation formula seems to give better accuracy. Exercise Explain why should give a reasonable approximation of f?(x) when f?(x) exists and h is small.

Problem 49A Demand Suppose the demand for a certain item is given by D(p) = ?2p2 ? 4p + 300, where prepresents the price of the item in dollars. a. Find the rate of change of demand with respect to price. ________________ b. Find and interpret the rate of change of demand when the price is $10.

Problem 50A Profit The profit (in thousands of dollars) from the expenditure of x thousand dollars on advertising is given by P(x) = 1000 + 32x ? 2x2. Find the marginal profit at the following expenditures. In each case, decide whether the firm should increase the expenditure. a. $8000 ________________ b. $6000 ________________ c. $12,000 ________________ d. $20,000

Problem 51A Revenue The revenue in dollars generated from the sale of x picnic tables is given by . a. Find the marginal revenue when 1000 tables are sold. ________________ b. Estimate the revenue from the sale of the 1001st table by finding R?(1000). ________________ c. Determine the actual revenue from the sale of the 1001st table. ________________ d. Compare your answers for parts b and c. What do you find?

Problem 53A Social Security Assets The table gives actual and projected year-end assets in Social Security trust funds, in trillions of current dollars, where Year represents the number of years since 2000. Source: Social Security Administration. The polynomial function defined by f(x) = 0.0000329x3 ? 0.00450x2 + 0.0613x + 2.34 models the data quite well. a. To verify the fit of the model, find f(10), f(20), and f(30). ________________ b. Use a graphing calculator with a command such as nDeriv to find the slope of the tangent line to the graph of f at the following x-values: 0, 10, 20, 30, and 35. ________________ c. Use your results in part b to describe the graph of f and interpret the corresponding changes in Social Security assets. Year Trillions of Dollars 10 2.45 20 2.34 30 1.03 40 ?0.57 50 ?1.86

Problem 52A Cost The cost in dollars of producing x tacos is C(x) = ?0.00375x2 + 1.5x + 1000, for 0 ?x ? 180. a. Find the marginal cost. ________________ b. Find and interpret the marginal cost at a production level of 100 tacos. ________________ c. Find the exact cost to produce the 101st taco. ________________ d. Compare the answers to parts b and c. How are they related? ________________ e. Show that whenever C(x) = ax2 + bx + c, [C(x + 1) ? C(x)] ? C?(x) = a. Source: The College Mathematics Journal. ________________ f. Show that whenever C(x) = ax2 + bx + c, .

Problem 55A Shellfish Population In one research study, the population of a certain shellfish in an area at time t was closely approximated by the following graph. Estimate and interpret the derivative at each of the marked points.

Problem 56A Eating Behavior The eating behavior of a typical human during a meal can be described by I(t) = 27 + 72t ? 1.5t2, where t is the number of minutes since the meal began, and I(t) represents the amount (in grams) that the person has eaten at time t. Source: Appetite. a. Find the rate of change of the intake of food for this particular person 5 minutes into a meal and interpret. ________________ b. Verify that the rate in which food is consumed is zero 24 minutes after the meal starts. ________________ c. Comment on the assumptions and usefulness of this function after 24 minutes. Given this fact, determine a logical domain for this function.

Problem 54A Flight Speed Many birds, such as cockatiels or the Arctic terns shown below, have flight muscles whose expenditure of power varies with the flight speed in a manner similar to the graph shown in the next column. The horizontal axis of the graph shows flight speed in meters per second, and the vertical axis shows power in watts per kilogram. Source: Biolog-e: The Undergraduate Bioscience Research Journal. a. The speed Vmp minimizes energy costs per unit of time. What is the slope of the line tangent to the curve at the point corresponding to Vmp? What is the physical significance of the slope at that point? ________________ b. The speed Vmr minimizes the energy costs per unit of distance covered. Estimate the slope of the curve at the point corresponding to Vmr. Give the significance of the slope at that point. ________________ c. By looking at the shape of the curve, describe how the power level decreases and increases for various speeds. ________________ d. Notice that the slope of the lines found in parts a and b represents the power divided by speed. Power is measured in energy per unit time per unit weight of the bird, and speed is distance per unit time, so the slope represents energy per unit distance per unit weight of the bird. If a line is drawn from the origin to a point on the graph, at which point is the slope of the line (representing energy per unit distance per unit weight of the bird) smallest? How does this compare with your answers to parts a and b?

Problem 57A Quality Control of Cheese It is often difficult to evaluate the quality of products that undergo a ripening or maturation process. Researchers have successfully used ultrasonic velocity to determine the maturation time of Mahon cheese. The age can be determined by M(v) = 0.0312443v2 ? 101.39v + 82,264, v ? 1620, where M(v) is the estimated age of the cheese (in days) for a velocity v (m per second). Source: Journal of Food Science. a. If Mahon cheese ripens in 150 days, determine the velocity of the ultrasound that one would expect to measure. (Hint: Set M(v) = 150 and solve for v.) ________________ b. Determine the derivative of this function when v = 1700 m per second and interpret.

Problem 58A Temperature The graph shows the temperature in degrees Celsius as a function of the altitude hin feet when an inversion layer is over Southern California. (See Exercise in the previous section.) Estimate and interpret the derivatives of T(h) at the marked points. Exercise In Exercise, tell which graph, a or b, represents velocity and which represents distance from a starting point. (Hint: Consider where the derivative is zero, positive, or negative.) a. b.

Problem 59A Oven Temperature The graph in the next column shows the temperature one Christmas day of a Heat-Kit Bakeoven, a wood- burning oven for baking. Source: Heatkit.com. The oven was lit at 8:30 a.m. Let T(x) be the temperature x hours after 8:30 a.m. a. Find and interpret T?(0.5). ________________ b. Find and interpret T?(3). ________________ c. Find and interpret T?(4). ________________ d. At a certain time a Christmas turkey was put into the oven, causing the oven temperature to drop. Estimate when the turkey was put into the oven.

Problem 60A Baseball The graph shows how the velocity of a baseball that was traveling at 40 miles per hour when it was hit by a Little League baseball player varies with respect to the weight of the bat. Source: Biological Cybernetics. a. Estimate and interpret the derivative for a 16-oz and 25-oz bat. ________________ b. What is the optimal bat weight for this player?

Problem 61A Baseball The graph shows how the velocity of a baseball that was traveling at 90 miles per hour when it was hit by a Major League baseball player varies with respect to the weight of the bat. Source: Biological Cybernetics. a. Estimate and interpret the derivative for a 40-oz and 30-oz bat. ________________ b. What is the optimal bat weight for this player?

Problem 1E By considering, but not calculating, the slope of the tangent line, give the derivative of the following. a.f(x) = 5 ________________ b.f(x) = x ________________ c.f(x) = ?x ________________ d. The line x = 3 ________________ e. The line y = mx + b