MTH 151
Exam 2
Fall 2012
Show all work in a neat and organized fashion.
Clearly indicate your answers.
100 points possible.
x=
−b±
√
b2−4ac
2a
1. (12 pts.) Use the Candidates Test (i.e., Closed Interval Method), showing all work, to find the
absolute maximum and absolute minimum values of f on the given interval.
f(x)=x3+7x2−5x, [−6,4]
2. (24 pts.) Differentiate each function. You do not have to simplify your answers.
(a)
y=√x sinx
(b)
y=
x5−1
x5+1
(c)
y=(x2+x4)6
(d)
y=tan2(cosx)
3. (12 pts.) (a) Somebody used the product rule and got this. Simplify it.
(Just simplify. Do not take the derivative.)
3(2x−1)2(2)(x+3)1/2+(2x−1)3(
1
2
)(x+3)−1/2
(b) Somebody used the quotient rule and got this. Simplify it.
(Just simplify. Do not take the derivative.)
3(x+2)2(x−3)2−(x+2)3(2)(x−3)
(x−3)4
4. (6 pts.) Find the limit. Neatly show work that justifies the answer, using only methods we have studied.
lim
x→0
sin4x
sin9x
5. (12 pts.)
Consider
y=4x−3x2.
(a) Find dy.
(b) Evaluate dy when x=5 and dx=0.8.
(c) Compute ∆y when x=5 and ∆x=0.8.
6. (12 pts.) Use implicit differentiation to find y′ and y".
x5+y5=1
(You don't have to simplify. You do have to express your answers
in terms of x and y.)
7. (12 pts.) A particle moves according to a law of motion s=f(t), where t is measured in seconds and s in feet.
f(t)=−sin
⎛ ⎝
πt
4
⎞ ⎠
, 0 ≤ t ≤ 10
Important : A supplemental page shows the graphs of f(t), f′(t), and f"(t).
You may use the graphs to answer parts (c), (d), and (e).
(a) Find the velocity at time t.
(b) What is the velocity after 3 seconds?
(Decimal approximation to 3 decimal places or exact, either way.)
(c) For 0 ≤ t ≤ 10, when is the particle at rest?
(d) For 0 ≤ t ≤ 10, when is the particle moving in the positive direction.
(e) For 0 ≤ t ≤ 10, when is the particle speeding up? slowing down?
8. (12 pts.) Westport is 3 miles west of Easthaven. Jay leaves Westport, traveling north
at 5 mi/hr. At the same time, Kay leaves Easthaven, traveling south at 4 mi/hr.
After 0.5 hr, at what rate is the distance between Jay and Kay increasing?
(Suggestion: in your drawing, add auxiliary lines to obtain a single, larger right triangle.)