Math 1220 Section 5
William Malone
Review 1
Name:
Below is a series of problems that are harder than the problems that will appear
on the test. The test will be roughly 12-14 questions long with problems of varying
complexity. You need to know all derivatives and integrals of common functions
discussed in class as well as the two double angle formulas and the pythagorean
identities. Any other reduction formulas or trig identities needed will be given on the
test.
1. Use logarithmic differentiation to find f 0 (x) where
√
(x3 − 6x + 9) 5 x7 − 5x3 + 2
√
.
f (x) =
3
x2 − 1
2. Let f (x) = cosh(x)arctan(x
2 +3)
find f 0 (x).
3. Find the derivative with respect to x of all of the following functions.
(a) f (x) = 4sinh(x)
(b) g(x) = log8 (tan(x2 + 1))
(c) h(x) = arcsin(x3 + cos(x))
(d) t(x) = arctan(sin2 (x) + 2x )
4. Find the function y(x) that satisfies the initial condition y(2) = 0 and satisfies
the differential equation
1
dy
=
.
2
dx
(3y + 6y + 3)(x2 − 3)
5. Find the a general solution to the differential equation y 0 + 4x3 y = x7 .
R
5 (x)
6. Find arctan
dx.
1+x2
R
7. Find cosh(x)ex dx.
R
8. Find x arcsin(x)dx.
R
9. Find log11 (3x)dx.
R x3
10. Find √1−x
2 dx
R
11. Find sin3 (x) cos7 (x)dx
12. Find
R
sin(3x) sin(7x)dx using the formula
1
sin(mx) sin(nx) = − [cos(m + n)x − cos(m − n)x].
2
13. Find
Z
14. Find
x4 − x3 − 6x2 + x + 2
dx.
x3 − 2x2 − 5x + 6
x−4
dx.
+ 2x + 2
Z
x2
15. Find
Z
R
sec6 (x)dx using the reduction formula
1
n−2
sec (x)dx =
secn−2 (x) tan(x) +
n−1
n−1
n
16. Find
x5 − sin(6x)
x→0
tan(4x)
lim
17. Find
e8x − e−5x
x→∞ e7x + e−5x
lim
18. Find
lim
x→∞
19. Find
R∞
0
1
1+
x
x
xe−2x dx
20. Find
Z
0
1
1
√ dx
x
.
Z
secn−2 (x)dx if n 6= −1.