MAT 151 Review for Test 4
1) Given the function f(x) = x4 – 4x3 + 10 find the following:
(You may not use the calculator.)
a) State the domain.
b) Find the x- and y-intercepts.
c) Check Symmetry, if any.
d) Find horizontal and vertical asymptotes, if any.
e) Find intervals of increase/decrease and local maximum and/or minimum values, if any.
f) Find concavity and inflection points, if any.
g) Sketch the graph of the graph.
2) Given the following function f(x) =
x2
x2  4
(You may not use the calculator.)
a) State the domain.
b) Find the x- and y-intercepts.
c) Check symmetry, if any.
d) Horizontal and vertical asymptotes.
e) Find intervals of increase/decrease and local maximum and/or minimum values.
f) Find concavity and inflection points, if any.
g) Sketch the graph of the curve.
3) Prove that the equation 4x5 + 3x3 + 3x – 2 = 0 has exactly one real root.
4) Given f(x) = x3 – 5x2 – 3x, verify that the hypotheses of the Mean Value Theorem are satisfied for a = 1 and
b = 3. Then find all number(s) c in the open interval (1, 3) that satisfy the conclusion of the MVT.
5) Find the number in the interval [0, 1] such that the difference between the number and its square is a
maximum.
6) You have been asked to design a 1 liter oil can shaped like a right circular cylinder. What dimensions will
use the least amount of material (i.e. find r and h)?
Note: r (radius) and h (height) are measured in centimeters, so 1 L = 1000cm3.
Volume : V = πr2h
Surface Area = 2πr2 + 2πrh
Evaluate the following limits. For each one state the indeterminate form.
ln( 2  e x )
7) lim
x  
3x
10)
lim
 
x  
2

sec x
1  tan x
8) lim (e  x)
x
1
x
x 0
11) lim
x 1
1
1

ln x x  1
12) Find the function f if f’’(x) = 1 – 6x + 48x2 and f(0) = 1 and f’(0) = 2.
9) lim
x 0
sin x
x
13) Find the function f if f’’(x) = 24x2 + 2x + 10 and f(1) = 5 and f’(1) = −3.
14) A particle’s acceleration along a straight line is given as a(t) = 6t + 4. It’s initial velocity is v(0) = −6cm/s
and its initial displacement is s(0) = 9cm. Find its position at t = 5 seconds.
15) Use Newton’s Method to find all roots of the equation sin x = x2 – 3x + 1 correct to six decimal places.
Note: You will only need to do the 3rd approximation, ie Find x3.