P A (Calculus )dx Semester Review Name Period Directions: Solve the following problems. Show work when necessary. Put the best answer in the blank provided, if appropriate.. Let y = g(x) be a function that measures the water depth in a pool x minutes after the pool begins to fill. Then g (5) represents: I. The rate at which the depth is increasing 5 minutes after the pool starts to fill. II. The average rate at which the depth changes over the first 5 minutes. III. The slope of the graph of g at the point where x = 5. (A) I only II only (C) III only (D) I and II (E) I and III (F) I, II, and III. The function y = f(x) measures the fish population in Blue Lake at time x, where x is measured in years since January, 950. If f (5) = 500, it means that (A) there are 500 fish in the lake in 975. there are 500 more fish in 975 than there were in 950. (C) on the average, the fish population increased by 500 per year over the first 5 years following 950. (D) on January, 975, the fish population was growing at a rate of 500 fish per year. (E) none of the above.. Let f be a function with f (5) = 8. Which of the following statements is true? (A) f must be continuous at x = 5. f is definitely not continuous at x = 5. (C) There is not enough information to determine whether or not f(x) is continuous at x = 5.. Suppose f is a function such that f (9) is undefined. Which of the following statements is true? (A) f must be continuous at x = 9. f is definitely not continuous at x = 9. (C) There is not enough information to determine whether or not f is continuous at x = 9.
5. Consider the function y = f(x) shown at right. (A) At what x-values is f discontinuous? At what x-values would this function not be differentiable? 6. Suppose that f is a function that is continuous at x = -. Which of the following statements is true? (A) f must be differentiable at x = -. f is definitely not differentiable at x = -. (C) There is not enough information to determine whether or not f(x) is differentiable at x = -. 7. Which of the following statements are always true? I. A function that is continuous at x = c must be differentiable at x = c. II. A function that is differentiable at x = c must be continuous at x = c. III. A function that is not continuous at x = c must not be differentiable IV. at x = c. A function that is not differentiable at x = c must not be continuous at x = c. (A) None of them I and III (C) II and IV (D) I and IV (E) II and III (F) I, II, III, and IV 8. Find all values of x at which the slope of the function f(x) = x is equal to -.
Directions: Questions 9 and 0 refer to the function y = f(x) shown in the figure. 9. At what x-value(s) within the domain of the graph is the slope of f approximately zero? (A) Never {-, } (C) {-, 0, } (D) {-0.5, 0.5} (E) {0} 0. At what x-values within the domain of the graph would the slope of f be undefined? (A) Never {-, } (C) {-, 0, } (D) {-0.5, 0.5} (E) {0}. For each of the following functions, I. State whether or not it is differentiable at x =. (How do you know?) II. If the function is differentiable, give an equation for the line tangent to the function at x =. (A) f(x) = x + f(x) = x (C) f(x) = x +, x x, x >. Consider the function f(x) = sin (kx) +. Given that f (0) = k, what is the approximate value of f(0.0)? (A) k + 0.0.0 (C).0k (D) k +.0 (E) 0.0k +
. Let f(x) = 6 x. Use the methods we did at the beginning of the year to find the instantaneous rate of change when x = -.. Consider the following table of values for the advertising budget of ACME Cola. Year Budget (A) Find the average rate of change for the period 99-00. Find the average rate of change for the period 996-00. (C) Find the average rate of change for the period 998-000. 99 996 998 000 00 00 8.9. 5.5 9.0.8 5.5 (D) Give an estimate for the instantaneous rate of change in 999. 5. Consider the function y = f(x) shown at right. Approximate the instantaneous rate of change at x =. (A) - - (C) 0 (D) (E)
Directions: Questions 6-7 relate to the function f shown in the figure. 6. Give the approximate value(s) of the x-intercepts of f. (A) f has no x-intercepts. {} (C) {-,, } (D) {0, } (E) {-, } 7. Over what interval(s) will the graph of f have negative values? (A) (-, ) (-, 0) (, ) ( ) (C) 0, (D) (-,-), (E) Never ( ) 8. Consider the function f(x) = x. On what intervals are the hypotheses of the Mean Value Theorem satisfied? (A) [0, ] [, 5] (C) [, 7] (D) None of these 9. Verify that the function f(x) = sin x satisfies the hypotheses of the Mean Value Theorem on the interval [, ]. Then approximate to decimal places all values of c in (, ) that satisfy the Mean Value Theorem equation. 0. Consider the following graph of f(x) = x sin x on the domain [-, ]. How many values of c in (-, ) appear to satisfy the Mean Value Theorem equation? (A) None One (C) Two (D) Three (E) Four or More
. At time t = 5, the rate at which the volume of a sphere V(t) is increasing is numerically equal to 8 times the rate at which its radius r(t) is increasing. Write an equation to match this statement.. The cost of operating the widget factory C(w) increases $ for every widget produced. Write an equation involving a derivative to describe the situation.. Let A(t) represent the deer population in a local forest preserve at time t years, when t 0. The population is increasing at a rate directly proportional to 00 A(t), where the constant of proportionality is k. Which of the following statements accurately reflects the situation? (A) A(t) = k[ 00 A(t) ] A(t) = k[ 00 A'(t) ] (C) A'(t) = k[ 00 A(t) ] (D) A'(t) = [ 00 ka(t) ] [ 00 A(t) ] (E) A'(t) = k. Consider the graph of f(x) = x sin x shown at right. Draw schematic diagrams (number line) to approximate where the first and second derivatives are positive, negative, and zero. (A) f f
(C) State the reasoning behind your diagrams in parts (A) and. How did you determine where each function would be positive, negative, or zero? Be specific. 5. Let the graph shown in the previous problem be g. (A) Estimate the intervals on which g(x) is increasing. Estimate the intervals on which g(x) is decreasing. (C) Estimate where g(x) has local extreme values. Directions: Use the following information to solve the following problems. f is continuous on [0, 8] and satisfies the following: x 0 x < x < 5 5 5 < x < 6 6 6 < x 8 f 0 + DNE 0 6. Based on this information, is there a point of inflection at x =? (A) Definitely Possibly (C) Definitely not 7. Based on this information, is there a point of inflection at x = 5? (A) Definitely Possibly (C) Definitely not 8. Based on this information, is there a point of inflection at x = 6? (A) Definitely Possibly (C) Definitely not
9. Shown at right is a graph of f and f. Sketch a possible graph of f on the same axes. 0. A function f(x) exists such that f (x) = ( x ) ( x + ). How many points of inflection does f(x) have? (A) None One (C) Two (D) Three (E) Unknown. Find all points of inflection of the function f(x) = x x + 6x 0. Directions: Use the following information to solve the following problems. Suppose f is continuous on [0, 6] and satisfies the following: x 0 5 f - - f 5 0-8 f - 5 DNE 6-0 x 0 < x < < x < 5 5 < x < 6 f + f +
. Identify all points of inflection. (A) There are no points of inflection. (5, -) only (C) (, ) and (6, -) (D) (, ) only (E) (, ), (6, -), and (5, -). Explain the reason for your choice the in the previous question.. The graph of g, a twice-differentiable function, is shown below. Choose the correct order for the values of g(), g (), and g (). (A) g() < g () < g () g () < g () < g() (C) g () < g() < g () (D) g () < g() < g () (E) Cannot be determined 5. Let g be a function defined and continuous on the closed interval [a, b]. If g has a local minimum at c where a < c < b, which of the following statements must be true? I. If g (c) exists, then g (c) = 0. II. g(c) < g(b) III. g is monotonic on [a, b]. (A) I only II only (C) III only (D) I and II only (E) I and III only 6. On the interval [-5, 5], f is continuous and differentiable. If f (x) = ( x ) ( x + ) x + (A) There is a local maximum on f at x = -. ( ), briefly explain the following conclusions.
There is a horizontal tangent but no extrema at x = -. (C) If f() = 7, then f() > 7. 7. From an 8 inch by 0 inch rectangular sheet of paper, squares of equal size will be cut from each corner. The flaps will then be folded up to form an opentopped box. Find the maximum possible volume of the box. 8. The diagonal of a square is increasing at a rate of inches per minute. When the area of the square is 8 square inches, how fast (in inches per minute) is the perimeter increasing? (A) 8 (C) (D) 6 (E) 6 9. (A) π A spherical snowball with diameter inches is removed from the freezer in June and begins melting uniformly such that it is shrinking cubic inches per minute. How fast (in square inches per minute) is its surface area decreasing when the radius is inch? (C) (D) (E) π π 0. Which is the slope of the line tangent to y + xy x = at (, )? (A) - 5 9 0 (C) (D) (E) 8 7 7
. (A) (D) If xy y = x 5, then dy dx = y x y x + 5 y xy y xy x x + y (E) y y xy (C) x 5 y y. Consider the curve given by x x y = y. (A) Show that dy x xy = dx x + y. Find all points on the curve where x =. Show there is a horizontal tangent to the curve at one of those points.. (A) - 5 The distance of a particle from its initial position is given by 9 s(t) = t 5 +, where s is feet and t is minutes. Find the velocity at t = ( t + ) minute in feet per minute. (C) (D) - 9 (E) - 7. The distance of a particle from its initial position is measured every 5 seconds and provided in the table below. Use the data to answer the questions that follow. Time (sec) 0 5 0 5 0 Distance (ft) 0 7 7 5 0 (A) Estimate the velocity of the particle at t = 5 seconds. Include units.
What is the average velocity on the time interval [5, 0] seconds? 5. The graph of the velocity of a particle is given below. On the same axes, draw the acceleration graph. 6. The number of liters of water remaining in a tank t minutes after the tank has started to drain is R( t) = t 0t 7t + 80. At what moment is the water draining the fastest? (A) 0 minutes minutes (C) (D) (E) minutes 5 minutes It drains at the same rate the whole time. 7. What is the slope of the graph of y = sinx at x = π? (A) - - (C) 0 (D) (E)
8. The area of a circle is A = π r. How does the instantaneous rate of change of the area with respect to the radius when r = compare to the average rate of change of the area as the radius changes from r = to r =? (A) The instantaneous rate of change is twice the average rate of change. The instantaneous rate of change is equal to the average rate of change. (C) The instantaneous rate of change is half the average rate of change. (D) The instantaneous rate of change is three times the average rate of change. (E) Their relationship cannot be determined from the given information. 9. Find the equation of the line tangent to the graph of y = x at the point on the curve where the y-coordinate is exactly one-third the values of the x-coordinate given x > 0. Show the work that leads to your answer. 50. Find the equation of the line tangent to y = x + x + at x =. (A) y = - 5 x + 9 y = 5 x (C) y = x (D) y = - x (E) y = - x + 7
Directions: Use the table below for solving the next two questions. x f(x) g(x) f (x) 7-5 g (x) - 5. (A) 5 The value of d ( dx f g ) at x = is: - (C) - (D) (E) 5. The value of d f dx g at x = is: (A) (C) 5 (D) 6 (E) 0 Directions: The function f and g and their derivatives have the following values at x = and x =. x f(x) g(x) f (x) g (x) 7-0 - 5. f(x) [ ] at x = (A) 0 (C) (D) 6 (E) None of These 5. (A) - f ( g(x) ) at x = 0 (C) (D) 9 (E)