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MA 125-8B, CALCULUS I
January 31, 2008
Name (Print last name first):..........................................
Student ID Number:...........................
TEST I
PART I
Part I consists of 7 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
Given that lim x→a f (x) = 5 and lim x→a h(x) = −2, find
lim x→a
3 h(x) 2 h(x) + f (x)
Question 2
If − 2 x ≤ g(x) ≤ −x^2 − 2 x + 1 for all x > 1, evaluate
lim x→ 1 +^
g(x)
if the limit exists.
Question 3
Evaluate the limit
lim x→ 2 π cos
3 π cos
( (^) x 2
Question 4
Find the limit
lim x→∞
−x^5 + 4x^2 8 x^5 − x^3 + 7
Question 5
Given that g(2) = −5 and g′(2) = 4, find an equation of the tangent line to the graph of y = g(x) at x = 2.
Question 6
Where is the function f (x) = 2|x − 7 | not differentiable? Question 7
Calculate the limit lim x→∞ tan−^1 (x − x^3 ).
Problem 2
Sketch the graph of an example of a function f such that
lim x→ 0 −^
f (x) = − 2 , lim x→ 0 +^
f (x) = 1, f (0) = 2,
lim x→ 2 −^
f (x) = 0, lim x→ 2 +^
f (x) = − 1 , f (2) = − 1.
Problem 3
(a) Evaluate the limit
xlim→ 3
x^2 − 2 x − 3 x^2 − 7 x + 12
(b) Evaluate the limit
lim x→ 9
x √ 1 + x
Problem 5
Let F (x) =
x
(a) Use the limit definition of the derivative to find F ′(2).
(b) Find an equation of the tangent line to the curve y =
x at the point (2, 3 .5).
