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MA 125-CV, CALCULUS I
February 22, 2007
Name (Print last name first):..........................................
Last Four Digits of Student ID:........................
TEST II
PART I
Part I consists of 6 questions. Clearly write your answer (only) in the space provided after each question. You do not need not to show your work for this part of the test. No partial credit is awarded for this part of the test!
Each question is worth 5 points.
Question 1
Differentiate the function y = 12 x^6 − 3 x^4 + x + 4.
Answer:.....................
Question 2
For what value of x does the parabola y = 3x^2 − 2 x have a horizontal tangent?
Answer:.....................
Question 3
Suppose h(x) = f (x)g(x) where f (2) = 3, g(2) = 4, f ′(2) = 2, and g′(2) = −6. Find the numerical value of h′(2).
Answer:..................
Question 4
Let h(x) =
cos x g(x) where g(0) = 2, g′(0) = 8. Find the numerical value of h′(0).
Answer:..................
Question 5
Let f (x) = h(g(x)) where g′(1) = 5, g(1) = −2, and h′(−2) = 3. Use the Chain Rule to find the numerical value of f ′(1).
Answer:..................
Question 6
If x = 4 − y^2 and dy dt = 5, find dx dt when y = 2.
Answer:..................
Problem 2
(a) Find the derivative of the function
F (z) =
z − 1 z + 1
(Simplify your answer!)
(b) Find the derivative of the function
y = tan (sin 2x).
Problem 3
(a) Use implicit differentiation to find an equation of the tangent line to the curve
x^2 y^2 + 4xy = 12y
at the point (2, 1).
(b) Find dy/dx by implicit differentiation when it is known that
y^2 + x sin y = 4.
Problem 5
A particle moves along the curve y =
1 + x^3. As it reaches the point (2, 3), the y-coordinate is increasing at a rate of 4 cm/s. How fast is the x-coordinate of the point changing at that instant?
