Math 112, Spring 2014, Solutions to Midterm I
1. You do not have to simplify your answers but make sure they are clearly written with properly used
parentheses where necessary.
√
6
(a) Find f 0 (x) if f (x) = 7x3 − 5 x + .
x
6
5
f 0 (x) = 21x2 − √ − 2
2 x x
d (x2 + 4)3
dx 4x − 7
d (x2 + 4)3
3(x2 + 4)2 · 2x · (4x − 7) − (x2 + 4)3 · 4
=
dx 4x − 7
(4x − 7)2
p
3
(c) Find the second derivative y 00 if y = 5x2 + 4
(b) Compute
1
(5x2 + 4)−2/3 (10x),
3
2. Let f (x) = −2x2 + 5x + 3.
y0 =
2
1
y 00 = − (5x2 + 4)−5/3 (10x)(10x) + (5x2 + 4)−2/3 (10)
9
3
f (x + h) − f (x)
using the f (x) given above. Simplify completely until there is no more
(a) Compute
h
h in the denominator.
−2(x + h)2 + 5(x + h) + 3 − (−2x2 + 5x + 3)
f (x + h) − f (x)
−4xh − 2h2 + 5h
=
=
= −4x−2h+5
h
h
h
(b) Take your answer above, make h → 0 and finish computing f 0 (x).
f 0 (x) = lim 4x − 2h + 5 = −4x + 5
h→0
(c) Find the equation of the tangent line to the graph of y = f (x) at the point where x = 1. Simplify
your answer. The slope of the line is f 0 (1) = −4 · 1 + 5 = 1. The point on the line is x = 1 and
y = f (1) = −2 + 5 + 3 = 6. So the line equation is
y − 6 = 1 · (x − 1)
which simplifies to
y =x+5
3. The Total Cost and Total Revenue for producing and selling Cosas are given by
TC(x) = x3 − 6x2 + 16x + 4
TR(x) = −x3 + 14x2 + 19x
where x is in hundreds of Cosas and the Total Revuenue and Total Cost are in hundreds of dollars.
(a) What is the maximum profit? The profit is maximized when M P = 0 or M R = M C:
−3x2 + 28x + 19 = 3x2 − 12x + 16
or
−6x2 + 40x + 3 = 0
using the quadratic formula the roots are
p
−40 ± 402 − 4(−6)(3)
x=
≈ 6.74 or − 0.07.
2(−6)
The second one is negative so the profit is maximised at x = 6.74. The moximum profit is
T R(6.74) − T C(6.74) ≈ 312.4 hundred dollars
(b) Give the longest interval where Marginal Revenue and Marginal Cost are both increasing. MR is
increasing when M R0 = −6x + 28 > 0 which is when x < 14/3 ≈ 4.67.
MC is increasing when M C 0 = 6x − 12 > 0 which is when x > 2.
So both are increasing when 2 < x < 14/3.
1
4. Two Cars are initially in Seattle. The distance is measured towards North. So if you are 10 miles
North of Seattle, the distance is 10, if you are 10 miles South of Seattle, the distance is -10. The rate
of change of distance functions for the two cars is given, A0 (t) for Car A and B 0 (t) for Car B.
(a) Circle the correct answers: Initially Car A is going NORTH. Initially Car B is going SOUTH
(b) When is the distance between the two cars greatest? at about 52 min
(c) When is the distance of Car A to Seattle greatest? at 120 min
(d) Do either of the cars stop at any point? Which one and when? Yes, Car A stops at 120 min, Car
B stops at about 16 min.
B(60.2) − B(60)
from the graph. B 0 (60) ≈ 40.
(e) Estimate
0.2
2