Spring 2003, MTH Practice for Exam 3 4.2, 4.3, 4.4, 4.7, 4.8, 5.1, 5.2, 5.3, 5.4, 5.5, 6.1

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1 Spring 23, MTH 3 - Practice for Exam 3 4.2, 4.3, 4.4, 4.7, 4.8, 5., 5.2, 5.3, 5.4, 5.5, 6.. An object travels with a velocity function given by the following table. Assume the velocity is increasing. t(hr) v(mi/hr) Using 6 intervals of equal width, obtain upper and lower estimates of the distance travelled between t = and t = 4 hours. Give the units on your answer. 2. Let f(x) = 8x 3 + 6x. (a) Find all critical points of f(x). Determine which are local minima, which are local maxima, and which are neither. (b) Find values of x for which f (x) =. inflection points. Determine which of these are (c) Assume 2 x.75 and make a table of values of f(x) including all endpoints, local extreme points, and inflection points. Name both coordinates of the global minimum and global maximum over this interval. (d) Give a useful graph of the function using Xmin = 2, Xmax =.75, and appropriate values of Y min and Y max. Label each point from the table with its coordinates. 3. Let f(x) = x 3 3x 2 9x + 5. (a) Find all critical points of f(x). Determine which are local minima, which are local maxima, and which are neither. (b) Find values of x for which f (x) =. inflection points. Determine which of these are (c) Assume 2 x 4 and make a table of values of f(x) including all endpoints, local extreme points, and inflection points. Name both coordinates of the global minimum and global maximum over this interval. (d) Give a useful graph of the function using Xmin = 2, Xmax = 4, and appropriate values of Y min and Y max. Label each point from the table with its coordinates. 4. Paris showed that plant responses to phosphorus fertilizer was approximated by y =.57.47x x, where y is the yield and x the units of nitrogen. Find the number of units of nitrogen that maximizes the yield.

2 x y 5 2 Figure : Figure for problem number 6 5. Let f(x) = 3 sin(x)+x. Use the second derivative to determine the exact values of the inflection points of f(x) for values of x in the range x 2π. Be sure to justify your answer. 6. A graph of y = f(x) is given above. Using 4 rectangles of equal width to estimate the value of 5 f(x) dx. 7. Let f(x) = x + 2 x 2 2x + 3. (a) Graph f(x) to determine if is monotonic on the interval x? (b) What is the sign of x + 2 x 2 2x + 3 dx (c) Using the Riemann Sums program on your graphing calculator, compute the value of this definite integral, accurate to 2 decimal places, that is, use a large enough n so that LHS RHS <.. The Riemann Sums program is available from our course web page. Click on Programs for Calculators. Note: Monotonic means that the function is either nonincreasing or nondecreasing. 8. The rate at which the world s oil is being consumed is continuously increasing. Suppose the rate (in billions of barrels per year) is given by the function r = f(t), where t is measured in years and t = at the start of

3 (a) Write a definite integral which represents the total quantity of oil used between the start of 995 and the start of 2. (b) What are the units of the integral which you gave in part (a)? 9. Let f(x) = sin x. (a) Sketch graph of f(x) and shade in the area bounded by f(x), the x axis, the line x =, and the line x = 3π 2. (b) Express this area using definite integrals. (c) Using the area interpretation of the integral, find 2π sin x dx.. Some of the values of g(x) Are given in the following table. x g(x) Estimate 2 as accurately as possible. g(x) dx. Given that F (x) = x 2 cos(x) and f(x) = F (x) = x(2 cos x x sin x), use the Fundamental Theorem of Calculus to calculate the value of 2π x(2 cos x x sin x) dx π 2. The graph of the derivative, f (x), of a function f(x), is given below. Answer the following questions by providing the x coordinates of the points. (a) Find all critical points of the function f(x). (b) Find all local maximum points of the function f(x). (c) Find all inflection points of the function f(x). (d) Where do you think the global maximum occurs? Explain. 3. The graph of f(x) is given below. Estimate the value of the definite integral 5 f(x) dx by computing the left-hand sum and the right-hand sum using four subdivisions (n=4). Then use average of the values of the left-hand and right-hand sums to give a better approximation of this definite integral. 3

4 x Figure 2: Figure for problem number y x Figure 3: Figure for problem number 3 4

5 Figure 4: Figure for problem number 4 4. The graph of the function y = f(x) is given above. (a) Estimate the definite integral 3 (b) Estimate the definite integral 5 3 (c) Estimate the definite integral 5 f(x) dx. f(x) dx. f(x) dx. 5. Using logistic regression the total sales of a new product is modelled by P (t), where P (t) is given in thousands of units and t is in months since the product s release. P (t) = + Ce..75t At the end of the 5th month, the total sales reaches 75. (a) Calculate the value of C. (b) For what value of t will the total sales increase the fastest? (c) What is the maximum potential sales in the long run? 6. When a certain car was tested to see how fast it can go, the following velocity data was recorded. t (hr) v (mi/hr) (a) Using 4 intervals of equal width, calculate an underestimate for the distance travelled by the car between times t = and t = 3. (b) Using 3 intervals of equal width, calculate an overestimate for the distance travelled by the car between times t = 3 and t =

6 x Figure 5: Figure for problem number 9 7. Use the first and second derivative of f(x) = 2x x 3 to determine (a) the intervals where f is increasing and the intervals where f is decreasing. (b) The intervals where f is concave up and the intervals where f is concave down. (c) Find the inflection points. 8. Consider the surge function f(x) = πxe πx. Use the F Max function on your graphing calculator to estimate the value of the point where f(x) has a global maximum in the given domain, x. 9. The graph of the derivative, f (t) of a function f(t) is given above. From the graph, determine (a) Intervals where f(t) is increasing and intervals where f(t) is decreasing. (b) Points where f(t) has a local minimum or a local maximum. (c) Inflection points of f(t). Justify your answers. 2. The velocity of an object moving along a line is given by where t is in seconds and v is in m/s. v(t) =.25.25t 2 6

7 Figure 6: Figure for problem number 22 (a) Sketch a graph of the velocity against time and shade in the area corresponding to the distance travelled between times t = 5 and t =. Is it positive or negative? (b) Represent the distance traveled between times t = 5 and t = as a definite integral. (c) Estimate the integral to decimal place using a calculator program. That is, use a large enough value of n so that LHS RHS <. and say what value of n you used. 2. A business sells an item for p dollars. The demand equation, p = 34.2q, gives p as a function of q, the quantity sold. (a) Give the revenue as a function of q. R(q) = (b) If the cost function is C(q) =.25q, what is the profit as a function of q? π(q) = (c) Find the value of q that gives a maximum profit. 22. A manufacturer s cost of producing a product, C(q) is given by the graph above. The manufacturer will sell the product for $ each, regardless of the quantity sold, so that the revenue is R(q) = q. For which quantity is the profit a maximum? Mark the quantity on the q axis by q. 23. A curve representing the total number of people, P, infected with a virus has the form of a logistic function. P = L + Ce kt where t is given in weeks since the first patients were infected. 7

8 .4 A.3.2 B. C x Figure 7: Figure for problem number 24 (a) It is estimated in that in the long run, approximately,, people will become infected. What is L? (b) Suppose that initially, people had the virus. That is, when t =, P =. Find C. (c) Assume k =, graph the function. Mark the inflection point and asymptote. (d) Using the trace function, find the smallest value of t for which P (t), <. 24. The following graph shows drug concentration curves after oral adminstration of 3 different related drugs. All drugs meet the current USP standards of potency, disintegration time, and dissolution rate. (a) give possible values for the minimum effective concentration and maximum safe concentration that would make products A and B the preferred drug. (b) Give possible values for minimum effective concentration and maximum safe concentration that would make product C the preferred drug. 25. Let F (x) = 3x x 2 and F () =. Calculate F (b) for b =,.5,,.5, 2, The graph of the derivative g (x) of a function g(x) is shown in the figure below. Suppose g() =. Find g(b) for b =, 2, A graph of the function f(x) is given below. Estimate the average value of f(x) between x = and x = The quantity of a radioactive substance at time t is given by Q(t) = 4(.96) t. Find the average value of Q(t) over the interval t 2. 8

9 Figure 8: Figure for problem number y x 2 Figure 9: Figure for problem number 27 9