Calculus IB Final Review
Name _________________________
6
11
14 16
17
1) Find each limit algebraically.
⎛ sin(x) ⎞
a. lim ⎜ 2
⎟
x →0⎝ x + 3x ⎠
b.
€
a. __________
⎛ 9x ⎞
⎟⎟
lim ⎜⎜
x → ∞⎝ x 2 + x ⎠
b. __________
c.
€
⎛ 9x ⎞
⎟⎟
lim ⎜⎜
x →−∞⎝ x 2 + x ⎠
c. __________
2) Find the first and second derivatives of the functions.
€
8
a. f(x) = 4 x + 7x 3 − 9x + 2
x
€b. g(x) = tan(x) + sec(x)
€
f ’(x) = ______________________________
f “(x) = ______________________________
g’(x) = ______________________________
g“(x) = ______________________________
3) Find any relative extrema using the first derivative test and the second derivative test.
x2 + 3
f(x) =
x
______________________________
4) Find the indefinite integrals.
⎛
1⎞
a. ∫ ⎜tan(x)sec(x) − 3x 4 + x + 3 ⎟ dx
⎝
x ⎠
______________________________
€
b.
(x +1) 2
dx
x
∫
______________________________
€
5) Find the area of the region bounded by the curves. Write as a definite integral.
a. y = 0, y = 3x 2 − 4 x +1, x = -3, and x = 2.
__________
€
b. y = 0, y = 2sin(x) , x = 0, and x =
π
.
2
__________
€
6) Given
a.
∫
€
b
a
a
∫ a (9πesin(
€
€
€
€
∫
x 2 dx = 10,
4 x3
€
b.
c.
a
b
xdx = 7,
a
__________
a
∫ b x 2dx + ∫ b xdx + ∫ b dx
€b
∫ dx = -4, find
a
)dx
€
a
b
__________
€
∫ a (3x 2 − 6x + 2)dx
__________