Practice Test 3
MATH 2250
Instructor: Nguyen
1.
Below is the graph y = f 0 (x) of the DERIVATIVE of f (x). Use it to answer the following
y
3
2
1
-4
-2
2
4
-1
-2
questions.
-3
(a) Find where the function is increasing.
(b) Find where any local extrema occur, and identify them as local min or local max.
(c) Find where the function is concave down.
(d) Find where any inflection points are located.
(e) Sketch a possible graph of f (x).
x
√
2. Let f (x) = 23 x3/2 + 14 x + 1
(a) Find where the function is increasing.
(b) Find where any local extrema occur, and identify them as local min or local max.
(c) Find where the function is concave down.
(d) Find where any inflection points are located.
(e) Sketch a possible graph of f (x).
3. Find the following limits:
x100 − x90 − 2
x→1 x5 − x3 − 2
(a) lim
(b) lim
x→∞
ln(x)
x
(c) lim x ln(x)
x→0
(d) limπ
x→ 2
tan(x)
sec(x) + 1
1
(e) lim 1 +
x→0
x
x
(f) lim (ex + x)x
x→0
(g) lim+ x tan(π/2 − x)
x→ π2
4. For each of the folowing functions, find the absolut maximum and minimum on the specified
interval.
(a) f (x) = x ln(x) − 2x on [1, e3 ]
2
(b) g(x) = 3x 3 − x2 on [−1, 1]
5. Do the following:
(a) State the Mean Value Theorem
(b) For each of the following, if the Mean Value Theorem applies, find the point it guarentees, else,
explain why the Mean Value Theorem does not apply
2
(i) f (x) = x 3 − x2 on [−1, 1]
(ii) g(x) =
x2 − 9
on [−1, 1]
x2 + 1
6. Complete the following optimization problems
(a) You are designing a rectangular poster to contain 300 sq. in. of printing with a 3 in. margin
at the top and bottom and a 1 in. margin at each side. What overall dimensions will minimize
the amount of paper used?
(b) Find the volume of the largest right circular cone that can be inscribed in a sphere of radius 4.
(d) You are planning to make an open rectangular box from an 8-in.-by-15-in. piece of cardboard by
cutting congruent squares from the corners and folding up the sides. What are the dimensions
of the box of largest volume you can make this way, and what is its volume? Use the convention
that length is longer than the width.
√
(c) A right triangle whose hypotenuse is 6 m. long is revolved about one of its legs to generate
a right circular cone. Find the radius, height, and volume of the cone of greatest volume that
can be made this way.
7. A rectangular storage container with an open top is to have a volume of 10 m3. The length of this
base is twice the width. Material for the base costs $10 per square meter. Material for the sides costs
$6 per square meter. Let x, y, h be the width, the length, and the height of the container respectively.
a) Find the total cost C in term of x
b) Find dimensions the container in order to have such cheapest container?
8. Compute the following integrals
Z
1 1
2
e
]dx
• (i) [ + + √
+ sin(x) + √
5
π x
1 − x2
x3
√
√
Z
3t t − 10 t
• ii)
dt
2t2
Z 1
√
1
• (iii)
(x2 + x3 +
)dx
1+x
0
Z π/3
• (iv)
sec(t) tan(t)dt
0
Z x
d
3
• (v) Find
( (t4 + √
)dt
dx 0
s 1 − t2
Z x3
d
• (vi) Find
(
(sin(t5 ) + 1)dt
dx 0