FINAL EXAMINATION, MAT 00 December, 03 Write your solutions in a blue book. To receive full credit you must show all work. You are allowed to use an approved graphing calculator unless otherwise indicated. Simplify your answer when possible, but use the precise value rather than an approximation when you have a choice. (Example: If the actual answer is π, then write π, not 3.4.) The 5 problems are worth a total of 00 points. The time limit is hours [50 minutes]. Cell phones are strictly prohibited!. (0 points) Use the definition of the derivative to differentiate the following function. f(x) = 3 x. (7 points each) Find the exact value of each of the following limits. Write,, or does not exist if appropriate. It is particularly important to show your work on this problem. x 3x + (a) lim x x x (b) lim x x ( (c) lim x 0 x ) sin(x) 3. (8 points each) Differentiate the following functions (a) f(x) = x ln(x) (b) g(x) = e cos(5x) (c) h(x) = arctan(x) x 4. Evaluate the following integrals (a) (8 points) (b) (8 points) (c) (9 points) w + w dw sin(x) + cos(x) dx 7 x dx
5. (0 points) Let x + 4 x < f(x) = x x < 4 x x 4 (i) Sketch the graph of f on the interval [ 4, 4]. (ii) Find all values of x in [ 4, 4] at which f is not continuous. (iii) Find all values of x in ( 4, 4) at which f is not differentiable, i.e., for which f (x) does not exist. 6. (0 points) You are driving when you suddenly need to come to a complete stop. Suppose your initial rate was 88 feet per second (this is 60 mph). Starting at the instant you begin to apply the brakes, you decelerate at a constant rate of 0 feet per second per second (feet per second squared). State your answers in proper units. (a) How much time does it take you to come to a complete stop? (b) How far do you travel from the instant you begin applying the brakes until you come to a complete stop? 7. (0 points) A woman with a laser pointer is standing 0 feet away from a wall (this is the distance x in the picture below). She slowly moves the laser up the wall at a rate of foot per second. How fast is the angle θ between the laser beam and the horizontal changing when the laser is pointing 8 feet above the ground (that is, y = 8)? Your answer should be in radians per second.
8. (0 points) The graph of a function f(x) is shown below. Evaluate intervals below. (a) [a, b] = [ 4, ] (b) [a, b] = [, 0] (c) [a, b] = [ 4, 3] b a f(x) dx for the 9. (0 points) The fundamental frequency (or pitch) y measured in hertz (hz) of a 0 cm length of piano wire varies with the tension T holding the wire taut. Tension is measured in newtons (N). The relationship between y and T is expressed by the formula y = 00 0T.6 78π (a) Find the total change in frequency when the tension is increased from 000 N to 000 N. State your answer in proper units. (b) Find the average rate of change in frequency with respect to tension when the tension is increased from 000 N to 000 N. State your answer in proper units. (c) Find the instantaneous rate of change in frequency with respect to tension at 000 N. State your answer in proper units. 3
0. (0 points) A rectangle will be inscribed in the region bounded below by the x-axis, to the left by the y-axis, and beneath the curve y = x. The image below is an example. What is the largest possible area of such a rectangle?. (0 points) Sketch the graph of the function f(x) on the interval ( π, π) based on the following information. Label all maxima and minima, intervals of increase and decrease, points of inflection, concavity, and asymptotes. (i) f (x) = 3 cos x (ii) f (x) = 3 sin x cos x (iii) lim x π f(x) = (iv) lim f(x) = x π (v) lim x π (vi) f(0) = x π f(x) = lim f(x) = +. (0 points) Write down and evaluate a Riemann sum with 4 terms that gives an upper 9 e x estimate for dx. You may round your answer to the nearest 0.. x 4
3. (0 points) Find the equation of the tangent line to the curve x 3 x = (y 3 y) at the point (0, ). 4. (0 points) Let f(x) be a function such that (i) f(x) is continuous and differentiable for all x (ii) f (x) for all x (iii) f(0) = 0 What can you conclude about f(0)? 5. (0 points) The graph of a function f(x) is shown below. If g(x) is the anti-derivative of f(x), find (a) the subintervals of (0, 8) on which g(x) is increasing (b) the location of the local maxima of g(x) (x-coordinates) (c) the subintervals of (0, 8) on which g(x) is concave up 5