M106 Practice Exam 3

[Note: These problems appeared on my M106 exams in the fall of 1995.]

Instructions: Show all of your work and clearly explain your answers. This is particularly important on problems with a numerical answer, to allow the possibility of partial credit. No books are allowed during the exam, but you may use your calculator.

Problem [1] Suppose that f(x) is a differentiable function such that f(2)=3, f'(2)=5 and f'(3)=7.

(a) Find dy/dx when x = 2 and y = -1, given that yf(x) + xy³ = -5.
(b) Find the line tangent to the graph of y = x⁷ - 8x² + 35x - 40 at x = 0.

Problem [2] Suppose the graph below gives the derivative g(x) of some function y = G(x). $Graph goes here.$

(a) Circle each x-value on the x-axis in the graph for which the original function G(x) has a critical point.
(b) Put an X (for maX) in each of your critical point circles for which the critical point is a local maximum of the original function G(x).
(c) Put an N (for miN) in each of your critical point circles for which the critical point is a local minimum of the original function G(x).
(d) Put an i at each point of the graph for which the original function G(x) has a point of inflection.

Problem [3] Consider the limit as h approaches 0 of [ln(e+h)-1]/h.

(a) This limit is actually the definition of the derivative of some function f(x) at some value x = a. Determine f(x) and a.
(b) Use your answer to (a) to determine the limit exactly.
(c) Use L'Hopital's rule to determine the limit exactly a second way.

Problem [4] Let h(x) = x2^x.

(a) For what exact x-value is h(x) as small as possible? Show how you find the correct x-value and how you know h(x) has a minimum there.
(b) For what exact x-value or values in the range -2 <= x <= -1 is h(x) as large as possible? Show how you find the correct x-value and how you know for values in the given range that h(x) has a maximum there.

Problem [5] Farmer Brown wants to fence off three sides of a rectangular area; the fourth side runs along a river and does not need a fence. She wants there to be 5000 square yards of land in the rectangle. What are the dimensions of the rectangle needing the least amount of fencing?