S15 MATH 1225 – Test 2 30 March 2015 NAME:
CRN:
Use only methods from class. You must show work to receive credit.
1. (10 pts) Find the equation of the normal line to the curve y = x5 + xy 2 − 3 at the point (1, 2).
2. (16 pts) Find the derivatives of the following functions:
(a) (8 pts)
f (x) = e3x +
ln x
cos−1 (x)
+ log2 (x)
(b) (8 pts) Assuming x > 0,
u(x) = x3x+1
(Do not simplify)
3. (14 pts) The position of a particle is given by
s(t) = cos
πt
+π
4
between t = 0 and t = 8. The graphs of s(t) and s0 (t) are depicted below.
s(t)
s0(t)
1.0
0.5
0.0
0.5
1.0
0
1
2
3
4
5
6
7
(a) (4 pts) What is the total distance traveled by the particle over the given time interval?
(b) (6 pts) When is the particle speeding up?
(c) (4 pts) When does the particle have negative acceleration?
8
√
4. (10 pts) Use linearization or differentials to approximate ln( 1.05).
5. (10 pts) Assume that the equation −5x + 2 cos(2x) = 0 has at least one real solution. Use a theorem from class to
show that the equation cannot have more than one real solution. Fully justify your answer.
6. (18 pts) A large balloon is rising at 20 ft/sec. The balloon is 10 ft above the ground at the point in time that the
back end of a car is directly below the bottom of the balloon. The car is traveling at 40 ft/sec. What is the rate of
change of the distance between the bottom of the balloon and the point on the ground directly below the back of the
car one second after the back of the car is directly below the balloon?
7. (6 pts) Suppose that Newton’s method is applied to a differentiable function f (x) using initial value x0 . What is the
method used to find?
8. (16 pts) Use the function f (x) = 13 x3 + 2x2 − 5x + 8 to answer the following:
(a) (6 pts) Find the interval(s) on which f is increasing.
(b) (4 pts) Find x values of any local extrema of f . Then classify each extremum as a local minimum or local
maximum.
(c) (6 pts) Find intervals of concavity of f and the x values of any inflection points.
Honor Pledge: I have neither given nor received aid on this exam. Signature: