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Main points of this exam paper are: Inflection Points, Graph, Derivative, Function, Interval, Increasing, Decreasing, Concave Up, Concave Down, Values
Typology: Exams
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FINAL EXAM - DECEMBER 14, 2010
Instruction: Read each question carefully. Explain ALL your work and give reasons to support your answers.
Advice: DON’T spend too much time on a single problem.
Problems Maximum Score Your Score
Total 150
1
1.(20 pts.) The graph of the derivative f ′^ of a function f over the interval [− 4 , 4] is shown.
f ’(x)
0 −1^1 2 3
−4 −3 −
(i) For what values of x is f increasing?
(ii) For what values of x is f decreasing?
(iii) For what values of x is f concave up?
(iv) For what values of x is f concave down?
(v) Find all the inflection points of f.
3.(16 pts.) The edges of a cube increase at a constant rate of 2cm/s. How fast is the volume of the cube changing when the volume is 125 cubic cm.?
x
x
4.(20 pts.) (i) Find an equation of the line tangent to the curve x^2 + xy − y^3 = 7 at the point (3, 2).
(ii) Is y = sin 2x a solution of the Initial Value Problem
y′′^ + 4y = 0 with y
(π 4
Justify your answer.
6.(20 pts.) The graphs of the functions f (solid graph) and g (dashed graph) are given below.
g (x)
2 f (x) 1 0 −1 1 2 3
− − − −
(i) Let a(x) = f (x) + g(x). Find a′(2). If it does not exist, explain.
(ii) Let b(x) = f (x)g(x). Find b′(1). If it does not exist, explain.
(iii) Let c(x) = f (g(x)). Find c′(0). If it does not exist, explain.
(iv) Let A(x) = f g^ ((xx)). Find A′(2). If it does not exist, explain.
7.(20 pts.) A rectangular garden (shaded region) with an area of 200 square meters is surrounded by a grass border of 1 meter wide on two sides and 2 meters wide on the other two sides (see figure). What dimensions of the flower garden minimize the combined area of the garden and borders?
2 m
1 m
flower garden
