Exam 2 Review
Fall 2011, Math 1210-007
Note: The following collection of problems illustrate the skills you will need for the exam, but is not meant to be an exhaustive list of problem types. Make sure you are familiar with problems done in class as well as assigned homework problems.
1. Determine the global maximum and minimum values of the following functions on the specified interval.
(a)
(b)
(c)
(d) f g
(
( t x ) =
) =
1
3 x t
2 x
3
+1
+ on
1
2 t h ( x ) = x (1 − x ) f ( x ) = x
8 x
2 +4 on
[ − 1 , 0]
2
− 6 t + 8 on [ − 4 , 0]
2
/
5 on [
1
2
, 2]
[0 , 3]
2. For each of the following functions, determine the intervals where f is increasing, decreasing, concave up, concave down, and find all points of inflection. Find all points that give a local maximum and all points that give a local minimum.
(a)
(b)
(c)
(d)
(e) g f r
(
(
( t x x
) =
) =
) =
1
3 x h ( s ) = s
4 t
3
3 x
3
( x − 2) 3
− 6 x
2
− 4 s
−
1
2
3 t
2
+ 9
+ 10 x + 3
− 6 t − 4 f ( θ ) = sin (2 θ ) − θ, −
π
4
≤ θ ≤
3. A farmer wishes to enclose a 4000 m 2
π
4 field and subdivide the field into four rectangular plots with fences parallel to one of the sides. What should the dimensions of the field be in order to minimize the amount of fencing required?
4. An open box is formed from a square sheet of cardboard by cutting equal squares from each corner and folding up the edges. If the dimensions of the cardboard are 18 cm by 18 cm, what should be the dimensions of the box so as to maximize the volume?
5. Suppose a rectangle has its lower base on the y = 8 − x
2 x -axis and upper vertices on the graph of the function
. Find the area of the largest such rectangle.
6. Sketch the graph of the following functions.
(a)
(b)
(c) f f
(
( x x
) = x
) = 4
8 x
2 x
+4
3
− x
4 f ( x ) = x
2
/
3
7. Refer to the graphs of is y = f ′ ( x ) below and complete the following: (i) Approximately on what intervals f increasing? Decreasing? Concave up? Concave down? (ii) Where are all local extrema located?
(iii) Sketch a graph of y = f ( x ) assuming that f (0) = 0 .
f’(x)
4 f’(x) 4
3
2
2
−1.0
−0.5
0.5
1.0
x
1.5
1
−2
−1.0
−0.5
0.5
x
1.0
−4 −1
(a) (b)
8. Let f ( x ) = x
3 on the interval [0 , 6] . Find all numbers c in (0 , 6) that satisfy the conclusion of the Mean
Value Theorem for derivatives.
9. Find the following antiderivatives.
Z
(a) (sin u + cos u − u ) d u
Z
(b) x
2
( x
3
+ 2)
6 d x

Z
(c) x cos (5 x
2
− 1) d x
(d)
Z 4 y p
3 y
2
− 24 d y
10. Find the particular solutions to the following differential equations satisfying the given conditions.
(a) d y d t d s
=
√ t ; y = 0 when t = 4
(b) = ( t
2
− 7) s
2
; s =
1
6 at t = 2
(c) d t d y d x
= xy
2
; y = 7 at x = 1
11. A partical is moving along a line with acceleration a ( t ) = 3 t
2
− 5 ft/sec 2 . Find the velocity and position functions if v
0
= − 5 and s
0
= 0 .
12. A rock is shot from a highly advanced slingshot straight upward from a point 5 ft above the ground with a velocity of 300 ft/sec. At what time will the rock reach its maximum height? (Recall that acceleration due to gravity is -32 ft/sec 2 ).
13. On the planet Gzyx, a ball dropped from a height of 20 ft hits the ground in 2 sec. If a ball is dropped from the top of a 200 ft-tall building on Gzyx, how long will it take to hit the ground? How does this compare to dropping the ball from a 200 ft-tall building on Earth?
14. If a woman has enough “spring” in her legs to jump vertically to a height of 2.25 ft on the earth, how high could she jump on the moon, where the surface gravitational acceleration is approximately 5.3
ft/sec 2 ?
15. Estimate the area under the curve y = 2 x + 5 over the interval [1 , 5] using a circumscribed polygon of
4 rectangles. Then use geometry to find the actual area.
16. Evaluate the following using any convenient method:
(a)
(b)
(c)
(d)
(e)
(f)
(g) d d d d d d x x y
Z x
(2 w − 5)
5
− 1
Z
0
(2 p
3
− 7)
5 x
Z −
1 d d w p
(cos 5 x − 3 x
2 y
) d x
Z x
3 d cos (3 w ) d w d x y
Z
3
4
2
Z
1 x
3
( z
2
0
Z
2 d x
+ 1) x p x
3
2 d z
+ 1 d x
0
(h)
(i)
Z h sec
2
2 x + x sec( x
2
)tan ( x
2
) i d x
(j)
Z
Z x
1 − x
2 d x cos x
(1 − sin x ) 2 d x
(k)
Z
1 x
2 p
4 + 5 x
3
0 d x
(l)
Z
π/ 9
π/ 18 sin 3 w d w
(m)
Z
π
π/ 2 sin
3
θ cos θ d θ