MATH 171.501
Practice problems for Examination 2
For questions 1 to 6 circle the correct answer.
e3x − 2x − 1
.
x→0
4x
1. Compute lim
(a) − 43
(b) 0
(c) − 12
(d) ∞
(e)
1
4
2. A bacterial culture starts with 1000 bacteria and after 6 hours there are 10000 bacteria.
How many bacteria are there after t hours?
(a) 10000 · 10t
(b) 1000 · 6t
(c) 1000 · 10t/6
(d) 106t
(e) 10000 · 106t
x
3. Find the derivative of the function f (x) = xe .
(a) xe
x (ln x+1/x)
1
x
(b) ex
(c) xe
+ ln x
x
x
x /x
x
1
x
(d) xe e2
(e) xe ex
+ ln x
1
4. Two cars start moving away from an intersection. Car A travels due south and its
distance from the intersection after t seconds is t2 + 2t meters. Car B travels due north
and its distance from the intersection after t seconds is 2t2 meters. How fast is the
distance between the two cars increasing when t = 1?
(a) 1 m/sec
(b) 4 m/sec
(c) 8 m/sec
(d) 5 m/sec
(e)
2
3
m/sec
2
5. Compute lim (1 + x2 )1/x .
x→0
(a) e
(b)
1
2
(c) 0
(d) 1
(e) 2
6. Let y be defined implicitly in terms of x by the equation −x3 cos(y) + y 3 = 2π 3 . Find
y 0 when (x, y) = (π, π).
(a) 2π 3
(b) 0
(c) −3π 2
(d) − 31
(e) −1
2
7. Let y be defined implicitly in terms of x by the equation tan−1 (xy + 1) = x. Find y 0
when (x, y) = ( π4 , 0).
(a)
8
π
(b)
π
4
(c) 2
(d) −2
(e) e
8. Given the function f (x) = x2 + ex + tan−1 (x), find (f −1 )0 (1).
(a) 0
(b)
1
2
(c) 1
(d) 2
(e) e +
5
2
9. Find the derivative of the function f (x) = tan−1 (ex ).
(a)
ex
1 + e2x
(b)
ex
1 + ex
(c)
1
1 + e2x
(d)
1
1 + 2ex
(e) ex tan−1 (ex )
3
10. (a) State what it means for a function f to be differentiable at a point x.
(b) Given the function f (x) = x2 , compute f 0 (x) directly from the definition of derivative.
11. A ladder of length 10 ft leans against a vertical wall. The bottom of the ladder slips
in such a way that after t seconds its distance from the wall is (4 + 2et−1 ) ft. How fast
is the top of the ladder sliding down the wall when t = 1?
12. Find the derivative of each of the following functions.
(a) f (x) =
(x + 2)8 (x − 1)3
√
(x + 1)17 x2 + 2
(b) f (x) = xsin 2x
(c) f (x) = sin(sin(x3 ))
√
13. Given the function f (x) = − cos( x + 1), find f 0 (16).
14. The position
of a particle at time t is given by the vector function r(t) =
√
2
4
hcos t, t + t i. Find the velocity at t = 1.
15. Find the second derivative of the function f (t) = t2 sin t.
4