Math 180, Lowman, Summer 2008, Old Exam Problems 1 Limit Problems

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1 Math 180, Lowman, Summer 2008, Old Exam Problems 1 Limit Problems 1. Find the limit of f(x) = (sin x) x x 3 as x Use L Hopital s Rule to calculate lim x 2 x 3 2x 2 x+2 x Given the function f(x) = (a) State the domain of the function. (b) Determine lim x 1 f(x). (c) Determine lim x 1 + f(x). { x 2 + x + 1 if x 1 3 x if x < 1 (d) Is f(x) continuous on its domain? Explain your answer. 4. Given the function { x f(x) = 2 + 4x + 5 if x > 2 x 2 4x 3 if x < 2 (a) State the domain of the function. (b) Determine lim x 2 f(x). (c) Determine lim x 2 + f(x). (d) Find the limit (two-sided) of f(x) at x = 2, if it exists. 2 Definition Of Derivative 5. Using the definition of derivative as the limit of the difference quotient, compute the derivative of f(x) = 3x 2 x. No credit will be given for use of the algebraic formula for the derivative. 6. Using the definition of derivative as the limit of the difference quotient, compute the derivative of f(x) = x No credit will be given for use of the algebraic formula for the derivative. 7. Let f(x) = x 3 + x. Calculate f (2) using the limit definition of the derivative. 8. Use the definition of the derivative as the limit of the difference quotient to find the first derivative of the function f(x) = 3 5 4x.

2 3 Find Derivative 9. Differentiate:(i) e x4 (ii) cos( ln x) (iii) e x sin(x 2 ). 10. Find the derivatives of the following functions. Please do not simplify your answers. (i)x x 2 + x (ii) x2 1 x (iii)ex cos x 11. Find dy dx when: (i) y = 1 + tan x(ii)y = ln(cos x) (iii)x 2 + 2y 2 = Find the derivatives of the following functions. Please do not simplify your answers. (i)x 3 x + x 2/3 2x (ii) x (iii)e x sin x 13. Find the derivative of the following functions. DO NOT SIMPLIFY. a) x 2 sin(x) + x b) (x x ) 99 c) ln(x 2 + 4) d) 1 e x 1 + e x 14. Find the derivative of the following functions. DO NOT SIMPLIFY. a) cos(x 5 ) + 1 x b) (x2 + 19x 1) x 3/2 c) 10 x ln(x 2 + 4) d) e 2x arctan(sin x) 15. Find the derivatives of the following functions. Simplify your answer a bit. (i)(x 2 2x+2)e x x 2 (ii)ln(cos x) (iii) x Find the derivatives of the following functions. Do not simplify your answer. (i)ln(sin x) (ii)cos(x 2 ) + 7 tan x (iii) e x2 1 + x 2 (iv)arcsin(2x) (v)ax2 e bx 17. Find h (x) when h(x) = (1 + x) 3 arctan(x). 18. Find g (x) when g(x) = x x Find f (x) when f(x) = ln(e x sin(3x)). 20. Find g (x) when g(x) = 8x2 5x+1 tan x. 21. Find f (x) when f(x) = e x2 cos(4x). 22. Find h (x) when h(x) = ln(x 3 6x + 2). 23. Find f (x) when f(x) = x 3 (x 1) 2.

3 24. Differentiate the following functions and show your work. (i) ln(x + x 3 )(ii) e cos( x) (iii) tan(3x 2). 25. Find the derivatives of the following functions. Please do not simplify your answers. (i)x x 5 + x x (ii) x (iii)e x sin x 26. Suppose f(x) = (1+cos x) x. Calculate f (2) to two decimal places. Explain what calculations you are making. Don t forget to set your calculator to radian mode.

4 4 Evaluate The Integral Problems 27. The temperature of a pie in a 350 oven is given by f(t) = e 0.1t where t is the time (in minutes) the pie has been in the oven. (i) Using an integral of f, write a formula for the average temperature of the pie during the first 40 minutes, 0 t 40. (ii) Calculate this average temperaute approximately. 28. (i) Find a function whose derivative is 3x 2. (ii) Evaluate: b 1 3x 2, dx 29. Evaluate: (i) b 30. Evaluate the integral: a dx (ii) x 1 x dx x dx. 31. Evaluate the integral: a 0 e x dx 32. Find the area above the x-axis, under the curve y = sin x, and between x = 0 and x = π. 33. Evaluate the integral: a 0 sin x dx. 34. Find the area above the x-axis, under the curve y = x 2, and between x = 0 and x = (i) Find a function whose derivative is 4x + 1. (ii) Evaluate: b 1 (4x + 1) dx.

5 5 Graph Problems 36. Sketch the graph of a function f(x), assuming the following information: f (x) = 0 only at x = 1, +1; while f (x) = 0 only at x = 1. f (x) and f (x) are undefined (infinite) only at x = 0. f( 1) = 1, f(1) = 2, while f(x) as x 0, and f(x) + as x 0 +. Between these points f (x) and f (x) have sign given by: x < 1 1 < x < 0 0 < x < 1 1 < x sign of f (x) + sign of f (x) Give rough sketches, for x 0, of the graphs of y = x 5, y = x, y = x 1/3, y = x 0, and y = x 2. Use a single coordinate system with x and y going from 0 to about 3 and label each curve. 38. Let h(x) = x a) Sketch h(x) on the interval from 0 to 2. b) Write out and compute the Left-Hand Sum and the Right-Hand Sum for h(x) with n=4. c) Sketch the Left-Hand Sum on the graph from part (a). 39. Let f(x) = x 4 4x Each answer must be fully justified by Calculus. Answers copied from a calculator are not correct. a) Find the critical points of y = f(x). b) Find the intervals on which f(x) is increasing, and on which f(x) is decreasing. c) Find the local maxima and local minima of f(x). d) Find the intervals on which the graph of y = f(x) is concave down, and concave up. e) Find the global maxima and minima of f(x) for 1 x 4. f) Carefully sketch the graph of y = f(x) for the interval 1 x 4. Include inflection points, and all points corresponding to critical points. 40. Let f(x) = x 4 4x 2. Each answer must be fully justified by Calculus. Answers copied from a calculator are not correct. a) Find the critical points of y = f(x) and the local maxima and local minima of f(x). b) Find the intervals on which f(x) is increasing, and on which f(x) is decreasing. c) Find the intervals on which the graph of y = f(x) is concave down, and concave up. d) Carefully sketch the graph of y = f(x) for the interval 1 x 4. Include inflection points, and all points corresponding to critical points. 41. Let h(x) = x.

6 a) Sketch h(x) on the interval from 0 to 2. b) Write out and compute the Left-Hand Sum and the Right-Hand Sum for h(x) with n=4. c) Sketch the Left-Hand Sum on the graph from part (a). 42. A function has the following properties: f is increasing. f is concave down. f(5) = 2. f (5) = 1/2. (i) Sketch a possible graph for f(x). (ii) How many zeros does f(x) have and where are they located?. (iii) Is it possible that f (1) = 1/4? Justify your answer. 43. Consider the function f(x) = x + 2 cos x, for 0 x 2π. (i) Find where f is decreasing. (ii) Find the largest and smallest values of f (either as decimals or exact). (iii) Find all points of inflection. (iv) Find where f is increasing most rapidly. (v) Sketch the graph of f The purpose of this problem is to study the graph of f(x) = for a > 0. (i) 1 + ae x Using your graphing calculator sketch the graph of f(x) for three different choices of a > 0. Describe the graphs: increasing and/or decreasing ranges, maxima or minima, concavity, and asymptotes. (ii) Calculate the derivatives f (x) and f (x) and find the x- and y-coordinate of the point of inflection. How does this point move as a increases? 45. Listed below are four functions, all of which take the value 0 at x = 0. sin x, x sin x, 1 + cos x, e x 1 Group these functions according to their tangent line approximations at 0. For each group sketch the graph of the tangent line and the functions which are tangent to it. 46. Let f(x) = x 3 + x 2 x 1. (i) Find and classify all the critical points of f. (ii) Find all the inflection points. (iii) Sketch the graph of f on the interval 1.5 x 1.5. (iv) Find the global maximum and minimum of f(x) for 1.5 x Let f(x) = x 3 x 2 x + 1. (i) Find and classify (as local max or local min) all the critical points of f. (ii) Find all the inflection points. (iii) Sketch the graph of f on the interval 2 x 2. (iv) Find the global maximum and minimum of f(x) for 2 x Let f(x) = x 3 + 2x 2 4x. (i) Find and classify (as local max or local min) all the critical points of f. (ii) Find all the inflection points. Justify your answer. (iii) Sketch the graph of f on the interval 3 x 3. (iv) Find the global maximum and minimum of f(x) for 3 x 3.

7 49. Let f(x) = 3x2 3 x 2 4. a) What are the zeros of f(x)? b) Find all asymptotes for f(x). c) Made a careful graph of y = f(x). Be sure to label the axes, label the zeros and indicate all asymptotes. 50. Give a formula for a function which has the following graph. 51. Give a formula for a function which has the following graph. y (0, 6) (10, 6). (5, 2) x 52. Sketch a graph of a function f(x) with the following properties: f(0) = 2

8 f (x) < 0 when x < 0 f (x) > 0 when 0 < x < 4 f (x) < 0 when 4 < x < 7 f (x) > 0 when x > 7 f(x) as x 7

9 6 Word Problems 53. If you jump out of an airplane at 5000 meters and your parachute fails to open, your downward velocity after t seconds will be approximately v(t) = 49(1 e 0.2t ) meters per second. Using an integral, estimate the distance that you will have fallen after 10 seconds. 54. You have $500 invested in a bank account earning 4.2% compounded annually. (a) Write an equation for the money M in your account after t years. (b) How long will it take to double your money? 55. The velocity v(t) of a car in miles per hour is given by the following graph a) How far does the car travel in the first hour? b) How far does the car travel between the times t = 1 and t = 5? Justify your answer explain how you calculate it. 56. Find the average value of f(x) = x between x = 4 and x = An open-top box is to be made with a square base and rectangular sides. Material for the box costs $.50 per square foot, and the final volume of the box should equal 32 cubic feet. For each of the following questions, you must show all your work to receive credit. a) Find a formula for C(x), the cost of the box, as a function of the length x of a side of the bottom of the box. b) What are the dimensions of the box to minimize the cost?

10 c) What is the minimum cost? 58. A ball is thrown straight up in the air. Its height above the ground is given by s(t) = t 16t 2. a) What is its initial velocity at time t = 0? b) Find the maximal height that the ball gets above the ground? 59. A family of rectangles has one corner on the curve y = x 2, base on the x-axis and its right side on the line x = 1. Find the dimensions of the rectangle with maximal area that fits this description. 60. A ball is thrown straight up in the air. Its height above the ground is given by s(t) = t 16t 2. a) What is its initial velocity at time t = 0? b) Find the maximal height that the ball gets above the ground? c) How fast is the ball going when it hits the ground? 61. A landscape architect plans to enclose a 4000 square-foot rectangle in a botanical garden. She will use shrubs costing $30 per foot along three sides and fencing costing $20 per foot along the fourth side. Find the dimensions that minimize the total cost. 62. If P dollars are invested at an annual rate of r%, then after t years this investment grows to F dollars, where ( F = P 1 + r ) t. 100 (i) Find df. (Assume P and t are constant.) dr (ii) What does the statement df = 160 tell you about the cost or benefit of an increase dr r=5 in the annual rate r? 63. A rancher wants to design a rectangular pasture with total area 15,000 square feet. The pasture will have a fence around the outside, and also a fence down the middle (subdividing it into two equal rectangles). What length and width of rectangle should he choose, in order to minimize the amount of fencing needed? 64. An open-topped box with a square base is to be constructed. The bottom of the box is to be made of gold-plated metal at a cost of $5.00 per square inch, and the sides of the box are to be made of silver-plated metal at a cost of $2.00 per square inch. The volume of the box must be 80 cubic inches. What are the dimensions of the cheapest box meeting these specifications? 65. The temperature of a pie in a 325 oven is given by f(t) = e 0.1t where t is the time (in minutes) the pie has been in the oven. (i) Write a formula using an integral for the average temperature of the pie during the first 30 minutes, 0 t 30. (ii) Calculate this average temperature with an error of at most 5.

11 66. Suppose that f(t ) is the cost in dollars per day to heat my house when the outside temperature is T degrees Fahrenheit. What does f (23) = 0.17 mean in practical terms? 67. A deposit of P 0 dollars is placed in a fund which earns compound interest. The amount P in the fund after t years is modeled by the exponential function P = P 0 e kt where P 0 and k are constants. You are given that after 4 years, the amount in the fund is $ , and after 5 years, the amount in the fund is $ (a) Determine the amount of the original investment P 0.(b) Find the amount in the fund after 10 years.

12 7 Table Problems 68. The function f(x) is increasing. Some of its values are given in the table: x f(x) (i) Compute the left and right Riemann sums with two subdivisions (n = 2) for the integral 2 1 f(x) dx. (ii) Explain what these sums tell you about the exact value of the integral 69. A car travelling on a gravel road had a flat tire and stopped unevenly. The table below shows its velocity (in feet per second) t seconds after the flat. t in sec v in ft/sec (i) Approximately how far did the car go in the first second? (ii) Approximately how much time did it take the car to stop? (iii) Approximately how far had the car gone when it came to a stop? 70. Given the following data about a function f, x f(x) (a) Estimate f (3.2). (b) Give the equation of the tangent line at x = One of the following tables of data is linear and one is exponential. Say which is which and give an equation which is a good fit for each table. 72. (a) 73. (b) x y x y The values of three functions f(x), g(x), h(x) are given in the table: x f(x) g(x) h(x) Which of the above functions is linear, which is concave up, and which is concave down? Justify your answer!

13 75. The following table gives the position s(t) of a particle at certain times t. Position is measured in miles and time in hours. t s(t) a) Estimate the velocity of the particle at times t = 1.4, 1.6 and 1.8. b) Use part a) to estimate the acceleration of the particle at time t = There is a function used by statisticians, called the error function, erf(x). Suppose from a table or a statistical calculator you find the following values: x erf(x) (i) Using this information, give an estimate for erf (0), the derivative of erf at x = 0. Only give as many decimal places as you feel reasonably sure of, and explain why you gave that many decimal places. (ii) With the extra information that erf(0.001) = could you refine the answer you gave in (i)? Explain. 77. The function f(x) is increasing. Some of its values are given in the table: x f(x) (i) Compute the left and right Riemann sums with two subdivisions (n = 2) for the integral 3 2 f(x) dx. (ii) How do these sums compare with the value of the integral? 78. A differentiable function f(x) has the following values given by the table x f(x) a) Using the data from this table, make a table of the approximate values of the derivative function f (x) for the same values of x. b) Is f(x) increasing or decreasing on the interval 2 x 8? Explain your answer. c) Using the data from your answer to part a), make a table of the approximate values of the second derivative function f (x) for the same values of x. d) Is f(x) linear, concave up, or concave down on the interval 2 x 8? Explain your answer.

14 79. Which of the following functions is linear, which is concave up, and which is concave down? Justify your choices. x f(x) g(x) h(x) The following table gives the number Q(t) of rabbits after t months. t Q(t) (i) Find a formula for Q(t) assuming exponential growth. population will reach (ii) Predict when the rabbit 81. 6) Determine which of three functions f(x), g(x), h(x) in the table below is: concave down; or concave up; or linear. Justify your answer x f(x) g(x) h(x)

15 8 Tangent Line Problems Write the equation of the line tangent to the graph of y = f(g(x)) at x = 2. You will need to know g(2) = 3, g (2) = 4, f(3) = 1, f (3) = Let f(x) be a function with f(3) = 2 and f (3) = 5. a) Give an equation of the line tangent to the graph of y = f(x) at x = 3. b) Estimate f(2.96). 84. Given the curve y 2 + 2y + e 3x = 4 a) Find dy/dx as a function of x and y, b) Give the equation of the tangent line to the curve at (x, y) = (0, 1). 85. Let f(x) = x 1/3. a) Give an equation of the line tangent to the graph of y = f(x) at x = b) Estimate f(997). c) Is your answer to part b) an over or under estimate? Justify your answer using calculus. 86. Let f(x) = cos x 0.5. (i) Find the equation of the tangent line to y = f(x) at x = π/3. (ii) Where does this tangent line intersect the x-axis? 87. Let f(x) = cos x 0.5. (i) Find the equation of the tangent line to y = f(x) at x = π/3. (ii) Where does this tangent line intersect the x-axis? 88. For the curve xy + 3y 2 = 18, (i) Find dy dx tangent line at the point (3, 2). in terms of x and y. (ii) Write the equation of the 89. Suppose that f(2) = 5 and f (2) = 3. Write the formula for the tangent line to f at x = 2. 9 Implicit Differentiation Problems 90. For the curve x 2 + 3y + y 2 = 19, (i) Find dy dx the tangent line at the point (1, 3). in terms of x and y. (ii) Write the equation of 91. Given the curve e xy + y 2 = 2 a) Find dy/dx as a function of x and y. b) Give the equation of the tangent line to the curve at (x, y) = (0, 1). 92. Find dy dx in terms of x and y when x and y are related by the equation 93. For the curve 3xy 2 x 2 y = 2 y 2 + x 2 y 7 = 0. (i) Use implicit differentiation to find dy dx. (ii) Find the tangent line at the point (x, y) = (1, 1).

16 94. For the curve x 2 2x + y 2 + 6y = 15 (i) Use implicit differentiation to find dy in terms of x and y. dx (ii) The point (x, y) = ( 2, 1) lies on the curve. Find an equation for the tangent line to the curve at this point. 10 Continuity 95. Find the value of k for which the function f(x) defined below is continuous for all values of x: f(x) = { kx + 15 if x < 2 3 kx 2 if x Approximation Problems 96. Let f(x) = x sin x. Using your calculator, estimate f (2). Explain in two or three sentences what calculations you are making. 97. Use the derivative approximation to f(x) = x at x = 9 to find the approximate value of f(2) = 3 and f (2) = 5. (a) Write the formula for the tangent line approximation to f at x = 2. (b) Estimate f(1.7).