COLUMBIA UNIVERSITY
CALCULUS I (MATH S1101X(3))
3RD SAMPLE MIDTERM 2 – JULY 12, 2012
INSTRUCTOR: DR. SANDRO FUSCO
Problem 1: (10 Points)
(a) What does it mean for
f to be differentiable at a ?
(b) What is the relation between the differentiability and continuity of a function?
Answer: 5 pts for (a), 5 pts for (b)
(a) A function f is differentiable at a number
if
f ′(a) exists.
a if its derivative f ′ exists at x = a ; that is,
(b) See Theorem 2.8.4. This theorem also tells us that if
f is not continuous at a , then if
f is not differentiable at a .
Problem 2: (10 Points)
(a) If
f ( x ) = 3 − 5 x , use the definition of a derivative to find f ′( x ) .
(b) State the domain of the function f and the domain of its derivative f ′ .
Answer: 6 pts for (a), 4 pts for (b): 2 pts for each domain
MATH S1101X(3)
IN-CLASS EXAM 2
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Problem 3: (10 Points)
Find the equations of the tangent line and normal line to the curve
P(0,2).
y = (2 + x ) ⋅ e − x at the point
Answer: 6 pts for finding the slope of tangent line, 2 pts for each equation
Problem 4: (20 Points)
Compute the derivatives of the following functions:
1.
g ( x) = x ⋅ e x
2.
f (v ) =
3.
f (t ) = sin(e t ) + e sin(t )
4.
f ( x) = ln sin 2 (x )
v 3 − 2v v
v
(
)
Answer: 5 pts for each derivative
1. By the Product Rule,
2. It is easier to simplify first, and then compute the derivative.
3. By the Chain Rule (applied twice):
MATH S1101X(3)
IN-CLASS EXAM 2
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Problem 5: (10 Points)
Prove that
(
)
d
1
−1
cos −1 ( x ) = −
. [Hint: cos ( x ) = y ⇔ cos( y ) = x and 0 ≤ y ≤ π ]
2
dx
1− x
Answer: 6 pts for setting up solution (implicit differentiation) correctly, 4 pts for correct proof.
Problem 6: (10 Points)
Find the equations of the tangent line and normal line to the curve
point P(2,1).
x 2 + 4 xy + y 2 = 13 at the
Answer: 6 pts for finding the slope of tangent line, 2 pts for each equation
Problem 7: (10 Points)
A tank is being filled with water at a rate of 2 ft3/min. The tank is in the shape of an inverted
cone (so the point of the cone is on the ground) with a radius of 2 ft and a height of 4 ft. How
fast is the depth of the water rising when the water in the tank is 3 ft deep?
Answer: 6 pts for setting up the solution correctly, 4 pts for correct solution
See Example 3 of Section 3.9 in the textbook (page 245 for 7th Ed., page 243 for 6th Ed.)
MATH S1101X(3)
IN-CLASS EXAM 2
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Problem 8: (10 Points)
1 
,2 . Find any absolute maximum or
 2 
Consider the function f ( x ) = x − ln( x ) having domain 
absolute minimum values of f (x ) , and find the x-values at which they occur.
Answer: 5 pts for critical numbers of the function, 5 pts for correct solution.
Problem 9: (10 Points)
(a) Verify that the function f ( x ) = x 3 − x 2 − 6 x + 2 satisfies the three hypotheses of
Rolle’s Theorem on the interval
(b) Find all numbers
[0,3] .
c that satisfy the conclusion of Rolle’s Theorem.
Answer: 5 pts for (a), 5 pts for (b)
MATH S1101X(3)
IN-CLASS EXAM 2
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