Download Sample Midterm II

MATH 113/114 Sample Midterm 2
Question 1. Find the derivative of the following functions
(DO NOT SIMPLIFY):
√
(a) f (x) =
x2 +2
x2 +3x
− sin(sin(sin 4x))
(b) f (x) = (2x + 1) · (x2 − 1) · sin(2x)
√
(c) f (x) = sin2 (tan(x3 + x))
Question 2. Use implicit differentiation to find an equation of the tangent
line to the curve
sin(x + y) = 2xy + 2x3
at the point (1, −1).
Question 3. Evaluate the limits:
sin(3x)
(a) lim
x→0 sin(2x)
x2 − 9
(b) lim
x→3 sin(x − 3)
Question 4. The volume of a right circular cone is V = πr2 h/3, where r is the
radius of the base and h is the height. Suppose that the height is increasing
at a rate of 1 cm/min while the volume of the cone is increasing at a rate of
40π cm2 /min. At what rate is the base of the cone changing when the height
is 3 cm and the volume is 100π cm2 .
Question 5. Use linear approximations to estimate the number sin(0.01).
Question 6. Find the critical numbers for the function f (x) = x3/4 − 2x1/4 .
Question 7. Find the absolute maximum and minimum values of the function
f (x) = 3x−4
x2 +1 on the interval [−2, 2].
x
Question 8. Verify that the function f (x) = x+1
satisfies the hypotheses of
Mean Value Theorem on the interval [1, 4]. Then find all numbers c that
satisfy the conclusion of Mean Value Theorem.
Question 9. Show that the equation 1 + 2x + x3 + 4x5 = 0 has exactly one
real root.
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