Calculus Review Final Exam 1.
2. ⎧⎪ 5 − x, x ≠ 2
Use the graph to find lim f (x) for f (x) = ⎨
x→2
⎩⎪ 7, x = 2
Name: ______________________________ Find the limit: lim(2x 2 − 5x + 3) x→5
3. x 2 − 6x + 9
x→ 3
x−3
Find the limit: lim
4. Find the value(s) of x for which f (x) =
removable or nonremovable. x+7
is discontinuous and label these discontinuities as x 2 − 49
5. 6. 7. Find the limit: lim−
x→0
3
2x
Use the graph to determine all x-­‐values at which the function is not differentiable. Find f '(x) : f (x) = 6x 5 − 8x 4 + 2x 2 − 78 8. Find f '(x) : f (x) =
9.
10. Find 11. Differentiate: y =
12.
Find 2x 3 − 3x 2
3
x
Find an equation for the tangent line to the graph of f (x) = 3x 2 + 6x + 5 at the point (2, 29). dy
: y = 2 cos x + 3sin x + x 2 dx
8x
4x 2 − 1
for y =
x−2
2x + 3
13. Find f ’(x) if f(x) = cos5 3x if x 2 y = 3x + y 2 14. Find 15. Find the second derivative of the function: f (x) = tan x 2x − 1
16. Find ALL critical numbers for the function: f (x) =
x−5
Evaluate the integral: ∫ 100 dx 17.
Evaluate the integral: ∫ (4x 4 + 3x 2 + 5x − 1)dx 18.
Evaluate the integral: ∫
19.
5
dm m8
Evaluate the integral: ∫ 8 x 3 dx 20.
21.
Evaluate the integral: ∫
2x + 3x 5 3
x3
dx cos 3 x
dx Evaluate the integral: ∫
1 − sin 2 x
22.
Evaluate the integral: ∫ 12 sec x tan x dx 23.
4
5
5
2
4
2
24. Given: ∫ f (x)dx = 12 and ∫ f (x)dx = 3 determine ∫ f (x)dx = ? 25.
Find the area of the region bounded by y = (x +2)2 + 3, the x-­‐axis, x = -­‐1, and x = 4. 5π 6
26.
∫
Evaluate: cos x dx π 3
4
27.
Evaluate: ∫
2
28. 3 + 4x + x 2
dx 3+ x
Evaluate: ∫ x 3 + 5x 2 dx 29. Evaluate: ∫ sec 2x tan 2x dx 30.
Evaluate: ∫ cos 3 4x sin 4x dx sin x
dx x
31.
Evaluate: ∫
32.
⎛ 2a 3 ⎞
Choose the expression equivalent to: ln ⎜ 2 ⎟ AND expanded COMPLETELY. ⎝ 9b ⎠
33.
Solve for x: ln(2x – 5) + ln (x + 3) = ln (x2 – 3) (Include ONLY real solutions!) 34.
Find for y = ln 3 3x 2 + 2 35.
Find the indefinite integral of: ∫
36.
Integrate: ∫
37.
Evaluate: ∫
38.
39.
40. 41.
1
dx 36 + x 2
3x − 5
dx x+2
Calculate the area of the region bounded by y = e3x, y = 0, x = 0, and x = 2. Identify the definite integral that represents the area of the region bounded by the graphs of and y = 4x − x 3 . Find the VOLUME of the solid bounded by the curves y = x 3 , y = 0, x = 0, and x = 1 rotated about the y-­‐axis. Select the correct integral to determine the volume of the solid generated by revolving the region bounded by the 2
graphs of the equations y = x + 2, y = 0, x = 0, and x = 2 the line x = 2. 1
dx x−5
42. Evaluate: ∫ x 3 sin xdx Find the indefinite integral using integration by parts: ∫ 3x 3e−2 x dx . 43.
44.
Evaluate the limit using L’Hopital’s Rule, if applicable. lim
x→0
sin x − x
2x 2
4x 2 − 3x + 2
Evaluate the limit using L’Hopital’s Rule, if applicable. lim
x→∞ 3x + 5x 2 + 7
45.
46.
Find the form of the partial fraction decomposition. Do NOT solve! 47.
Find the form of the partial fraction decomposition. Do NOT solve! 48.
49.
Integrate: ∫
50.
x+9
x (x 2 − 8)
2
Find the form of the partial fraction decomposition. Do NOT solve! −x + 5
dx 2x 2 + x − 1
Integrate: ∫
1
dx x −1
2
x −1
2x + x − 15
2
(
5x − 7
x + 3x + 1 ( x + 4 )
2
)