18. Find lim (e2x + x)1/x .
Math 151 Week in Review
Review For Exam 3 (4.3-5.7)
Monday Nov 29, 2010
Instructor: Jenn Whitfield
x→0
19. Using the graph of f ′ (x) below, determine all critical values of f (x), intervals
where f (x) is increasing and decreasing, local extrema of f (x), intervals where f (x)
is concave up and concave down, the xcoordinates of the inflection points. (Assume f (x) is continuous.)
Thanks to Amy Austin for contributing some problems.
All prolbems in this set are copywrited
1. Solve log(5 − x) + log(2 − x) = 1 for x.
2. Evaluate log 1 36.
6
3. Solve log4 (x2 − 16) − log4 (1 − 2x) = 1 for x.
4. If f (x) = 2 ln(arctan(x)), find f ′ (1).
y=f’(x)
5. Given f (x) = (cos x)tan x , find f ′ (x).
6. Let H(x) = ln (arcsin(x)).
1
3
5
7
9
(a) Determine the domain of H(x).
(b) Find H ′ (x).
(c) Find the domain of H ′ (x).
180◦
7. A bowl of soup at temperature
is
placed in a room that is kept at 70◦ F. If
the temperature of the soup is 150◦ F after
2 minutes, when will the soup be an eatable
100◦ ?
8. If 100 grams of a radioactive substance is
initially present and after 4 years 20 grams
remain, what is the half life of the substance? How much of the substance remains
after 2.5 years?
9. Find the value of tan sin
−1
2
.
5
10. Find the derivative of y = x2 sin−1 (e3x ).
11. Find the equation of the line tangent to y =
tan−1 (2x − 1) when x = 1.
12. Compute the exact
1 + 2x
.
lim arccos
x→∞
5 − 4x
−1
13. Compute sin
x→0
cos x − 1
.
x2
16. Find lim 2x cot x.
x→0+
17. lim
x→∞
2
1+
x
4x
of
4π
.
sin
3
14. Find
an
expression
tan(arccos x).
15. Find lim
value
equivalent
to
20. Find the absolute maximum and minimum
of the given functions on the given interval.
(a) x3 − 5x2 + 3 on [−1, 3]
(b) x ln x on [e−2 , 1]
21. Find the critical numbers for f (x) = x(4 −
x)2/3 .
22. Find the intervals where f (x) = (x − 2)2 e2x
is increasing and decreasing and identify all
local extrema.
1 5
x +
23. Find the intervals where g(x) =
20
1 4 1 3
x + x is concave up and concave down.
6
6
Identify all x-coordinates for the inflection
points.
24. An open-topped rectangular box is to have
a volume of 36000 cm3 . If its bottom is
a rectangle whose length is twice its width,
what dimensions would minimize the total
surface area of the box?
25. Find the dimensions of the isosceles triangle
of largest area that can be inscribed in a
circle of radius 2.