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Calculus 1 Final Exam B April 28, 1997 Instructions: Do not write your answers on this paper, use separate answer sheets. This is a closed-book test. To receive credit, you must show all work. You are not required to simplify your answers. 1. (10 pts.) Graph the equation y − 10 = −2(x − 2)2 . Is this the graph of a function y = f (x)? If yes, is the function y = f (x) invertible? 2. (10 pts.) A population triples in size every 15 days. The size of the population is initially P 0 = 135. (a) Give a formula for the size P = f (t) as a function of time. (b) Determine at what time the size is 300. Do not round oﬀ — leave your answer in terms of logarithms. 3. (10 pts.) For the graph below, give a function whose graph best matches the one given. y 10 5 x −4 −2 2 4 6 3 4. (10 pts.) Express the function w(x) = √ as a composition of the following functions. Use as x2 − 4 many functions from the list as possible. √ 1 f (x) = x g(x) = x2 h(x) = x − 4 j(x) = 3x k(x) = x 5. (10 pts.) The ﬁgure below is the graph of y = f (x). Sketch the graph of y = f (x). y 2 @ y = f (x) @ 1 @ @ x -2 -1 1 2 -1 -2 6. (10 pts.) The ﬁgure below gives the graph of distance traveled by a car over a 4 hour period. 150 miles 100 50 1 2 3 4 hours (a) Find the average velocity for the ﬁrst 4 hours. (b) At which time value is the velocity of the car greatest? (c) Give the time interval (or intervals) where the car is speeding up. (d) Give the time interval (or intervals) where the car is slowing down. f (x + h) − f (x) √ 7. (10 pts.) Find the derivative f (x) by calculating the limit lim for f (x) = x. h→0 h 8. (10 pts.) For sin x if x ≤ 0, f (x) = 2 A + (x − B) if x > 0 , where A and B are constants, (a) Find the left-hand and right-hand derivatives f− (0), f+ (0). (b) Find the left-hand and right-hand limits lim− f (x), lim+ f (x). x→0 x→0 (c) Find values for A and B such that f (x) is continuous and diﬀerentiable at x = 0. 1 2 9. (20 pts.) The table below gives the outputs that the functions f (x), g(x), f (x) and g (x) assign to the x values -2, -1, 0, 1, 2, and 3. Use this information to compute the derivatives indicated. x f (x) g(x) f (x) g (x) -2 3 -10 -3 -2 -1 -2 8 9 6 0 1 -3 4 -5 1 2 3 2 -4 2 -1 -13 7 2 3 0 21 5 -10 (a) F (−1), if F (x) = f (x)g(x). f (x) (b) F (2), if F (x) = . g(x) (c) F (1), if F (x) = ln (g(x)). (d) F (3), if F (y) = f −1 (y). 10. (10 pts. each) Find the derivatives of the following functions. (a) f (x) = ln(x + 2) Tan−1 4x sin 9y (b) g(y) = 5 √y +8 (c) h(t) = et − 7t dy (d) Find if x and y are related by the equation x2 y 2 = 2x − 3y + 7. dx 11. (30 pts.) The ﬁgure below is the graph of y = f (x). Assume y = f (x) is deﬁned on the domain [a, g]. y a b c d e f g x (a) Give the intervals where f (x) is concave up. (b) Give the intervals where f (x) is concave down. (c) Give the x values where f (x) has an inﬂection point. (d) Give the x values where f (x) has a critical point. (e) Give the intervals where f (x) is increasing. (f) Give the intervals where f (x) is decreasing. (g) Give the x values where f (x) has a local maximum. (h) Give the x values where f (x) has a local minimum. (i) Sketch a possible graph of f (x). 12. (30 pts.) The illumination I from a single light source is directly proportional to the strength S of the light source and inversely proportional to the square of the distance d from the source. If two lights are 45 feet apart and are rated at 125 and 175 units of strength respectively, at which point between them is the total illumination the least? The complete solution to this problem will include all of the following steps. (a) Give the objective function in terms of all of the variables. Deﬁne all variables. (b) Give the constraints. (c) Solve the constraint equation and eliminate all but one independent variable from the objective function. (d) Investigate the objective function at the endpoints and check for asymptotes. If possible, determine that the absolute extremum occurs in the interior of the domain. (e) Find the ﬁrst derivative and all remaining critical points. Locate the absolute extremum. (f) Give a complete answer to the question.