Diagnostic Problems
Math 1220-7
Due Friday, August 24, 2012
Key
Directions: Show all work for full credit. Clearly indicate all answers. Simplify all mathematical expressions completely. No calculators are allowed on
this assignment. This assignment is due by the end of class on Friday, August
24.
1. What was the last math class that you took? When did you take it?
2. What other math classes do you need to take?
3. What is your major?
4. Find the derivative of each of the following functions:
√
1
(a) f (x) = 3x2 + 2x + + 4 x3
x
f (x) = 3x2 + 2x + x−1 + 4x3/2
f 0 (x) = 6x + 2 − x−2 + 6x1/2
(b) h(t) = 2(4t + 5)5 sin(3t) + cos2 t
h0 (t) = 2 · 5(4t + 5)4 · 4 sin(3t) + 2(4t + 5)5 · 3 cos(3t) − 2 cos t sin t
= 40(4t + 5)4 sin(3t) + 6(4t + 5)5 cos(3t) − 2 cos t sin t
x3 − 4
(c) g(x) =
cos x
3x2 cos x + (x3 − 4) sin x
g (x) =
cos2 x
0
5. Calculate Dx y by implicit differentiation if x and y satisfy
xy + sin(xy) = 1.
xy 0 + y + cos(xy)(xy 0 + y) = 0
xy 0 + xy 0 cos(xy) = −y − y cos(xy)
y
−y(1 + cos(xy))
=−
y0 =
x(1 + cos(xy))
x
6. A child is flying a kite. If the kite is 60 feet above the child’s hand level
and the wind is blowing it away from the child on a horizontal course at
6 feet per second, how fast is the child playing out cord when 100 feet
of cord is out? Assume that the cord between the hand and the kite
remains straight.
b
Q
Q
Q
c QQ a = 60
Q
We want to find dc
dt when c = 100 (b = 80 by the Pythagorean Theorem).
db
We know that dt = 6 and da
dt = 0. By the Pythagorean Theorem,
a2 + b2 = c2
db
dc
2b
= 2c
dt
dt
dc b db
80
ft
= ·
=
· 6 = 4.8
dt
c dt
100
sec
7. Let f (x) = x3 − 3x. Find the function’s critical points and determine
whether they give a local maximum or local minimum by using either
the First Derivative Test or the Second Derivative Test. Also give the
maximum and minimum values.
f 0 (x) = 3x2 − 3 = 3(x − 1)(x + 1), so there are critical values at x = ±1.
Since f 0 (x) > 0 for x < −1 and x > 1, and f 0 (x) < 0 for −1 < x < 1,
there is a local maximum of 2 at x = −1 and a local minimum of −2 at
x = 1.
8. Evaluate the following integrals:
Z
1
(a)
x3 − 3x + 4 + 3 + sin x + cos x dx
x
Z
x3 − 3x + 4 + x−3 + sin x + cos x dx
=
Z
x sin(x2 ) dx
(b)
1
2
Z
1
4
(d)
1
Z
cos(x2 )
2x sin(x2 ) dx = −
+C
2
(t3 + 1)(t4 + 4t)5 dt
(c)
Z
1
x4 3x2
−
+ 4x − 2 − cos x + sin x + C
4
2
2x
7
Z
1
4(t3 + 1)(t4 + 4t)5 dt = (t4 + 4t)6 + C
24
1
dx
2x + 2
7
Z
√
√
1 7
1
2(2x + 2)−1/2 dx = · 2 (2x + 2)1/2 = 16 − 4 = 2
2 1
2
1
√
9. Find the function with the derivative f 0 (x) = sin x + 3 cos x + x2 such
that f (0) = 5.
Z
f (x) =
sin x + 3 cos x + x2 dx
x3
+C
= − cos x + 3 sin x +
3
5 = f (0) = −1 + C, so C = 6
x3
f (x) = − cos x + 3 sin x +
+6
3
10. Calculate the area between the curve y = sin x and the x-axis from x = 0
to x = 2π.
Z 2π
sin x dx
A=
Z 2π
Z0 π
(− sin x) dx
sin x dx +
=
π
0
π
2π
= − cos x + cos x
0
π
= 1 − (−1) + 1 − (−1)
=4