Calculus Tune-up: Math 104
Worksheet 1
Review of Derivatives and Integrals
Basic derivatives. Compute the derivatives of the following functions:
1. sin x
5. ex
2. cos x
6.
1
x
3. x3
7.
1
x2
4. 3x
8. xπ
Derivative rules. Compute the derivatives of the following functions:
9. 2 sin x + cos x
10. esin x
11.
x+3
x2 −x
12. tan x (Hint: tan x =
13. sec x (Hint: sec x =
14. xex
15. (2x + 1)9
16. sin(x) cos(x2 )
sin(x)
cos(x) )
1
cos(x) )
More advanced derivatives.
17. Find
dy
dx
as a function of x and y.
xy + y 2 + x2 = 3
18. Find
d
dx (arcsin(x)).
19. Show that
20. Find
d
dx (ln x)
d
g(x) ).
dx (f (x)
= x1 .
Basic anti-derivatives. Compute the following anti-derivatives:
1.
R
sin xdx
2.
R
cos xdx
3.
R
x3 dx
4.
R
dx
5.
R
ex dx
6.
R
1
dx
x2
7.
R
xπ dx
u-substitution. Use a u-substitution to compute the following anti-derivatives:
Z
8.
sin x cos xdx.
Z
x2 ex dx.
Z
x−1
dx.
x2 − 2x
Z
x(x2 + 1)3 dx.
9.
3
10.
11.
Definite integrals, and the fundamental theorem of calculus. Compute:
Z 4
12.
(x2 + 1)dx.
2
x
d
13.
dx
Z
d
14.
dx
Z
t sin(t )dt .
2
0
x2
!
t2
e dt .
x
Areas between curves.
14. Find the area between the curves y = 2x and y = x2 .
15. Find the area above y = cos(x) and below y = sin(x) between their first two intersections
(see diagram).