INTRO TO CALCULUS
REVIEW – FINAL EXAM
NAME:
DATE:
A. Equations of Lines (Review Chapter)
y = mx + b (Slope-Intercept Form)
Ax + By = C (Standard Form)
y – y1 = m(x – x1) (Point-Slope Form)
Problems:
1.
Find the equation of a line passing through point (5, -2) with slope ¾. (State your answer in slope-int. form)
2.
Find the equation of the line passing through the points (-1, 3) and (2, -4). (State your answer in standard form)
3.
Find an equation for a line parallel and an equation for a line perpendicular to the line 4x – 2y = 3 and passing
through the point (1, 4). Write your answers in standard form.
B. Solving Equations (Review Chapter)
1.) Quadratics – try to solve by factoring or quadratic formula.
2.) Rational Expressions – multiply each term by the LCD to eliminate fractions.
3.) Radical Equations – Isolate the radical first, then undo root by squaring both sides (
(3
), cubing both sides
), etc.
Problems:
Solve each equation for all real values of x.
4. 6x2 – x – 15 = 0
7.
2
x+2
+
4
x−3
=
5. 2x2 + 3x – 7 = 0
5x
6.
2x − 3 − 5 = 2
8. 23 3 x + 1 + 3 = 7
( x + 2)( x − 3)
C. Domain and Range (Review Chapter)
Domain – the set of “x”-values for a relation or function.
Range – the set of “y”-values for a relation or function.
Problems:
Determine the domain and range for each.
9. f ( x ) =
3
2x − 3
x +1
10. f ( x ) =
2x − 1
11. f ( x ) =
2
x − 3x + 1
2
x − 3x + 2
D. Limits (Chapter 1)
To evaluate limits try: 1.) Direct Substitution
2.) Test values that approach x.
3.) Try to factor and cancel any common factors.
Problems:
12.
3x + 5
13.
lim 5 x − 3
x→2
15.
lim+
x → −1
2
x − 2x + 1
16.
x +1
2
x −9
14.
lim x − 3
x →3
lim
x →0
1
17.
lim
x →1− x + 1
4+ x −2
x
x−3
lim
x →2 + x − 2
E. The Derivative (Chapter 2)
1.) Definition of the Derivative f ' ( x ) = lim
Δx→0
2.) Power Rule: f(x) = axn
5.) d
dx
(sin x ) = cos x
Δx
f ’(x) = naxn-1
3.) Product Rule: f(x) = g(x) · h(x)
4.) Quotient Rule: f(x) =
f ( x + Δx ) − f ( x )
g ( x)
h( x)
f ’(x) = g(x) · h’(x) + h(x) · g’(x)
f ’(x) =
h( x ) ⋅ g ' ( x ) − g ( x ) ⋅ h' ( x )
[h( x)]2
6.) d
(cos x ) = − sin x
dx
*In order for a function to have a derivative on any open interval (a, b), then the function must be continuous on (a, b)
and not have any sharp points on (a, b).
Ex: Function is not differentiable at x = 1
(Sharp point)
1
Function is not differentiable at x = 0 (Not continuous.)
Problems:
18. Find f ’(x) using the definition of the derivative. f(x) = x2 + 3
For #19 – 30, find the derivative using any method.
19. f ( x ) =
x−
1
x
20. f ( x) =
2x3 − 1
x2
21. f ( x ) =
x +1
x −1
22. f ( x ) = 1 − x 3
23. f(x) = (x2 – 1)(2x2 + x + 1)
24. f(x) = 3cos(2x – 1)
25. f(x) = 1 – 2sin x + 2 sin2 x
26. f(x) = x cos x
27. x2 + y3 = 10
28. x2 + xy + y2 = 16
29. y = cos3 (4x)
30. f (x) = x3 – 3x2
31. Find f ”(x) if f(x) = 2x2 + sin 2x
33. Find the equation of the line tangent to the curve f ( x) =
32. Find the slope of the line tangent to the curve
f(x) = (x – 1)(x2 – 2) at (0, 2).
x 2 + 2 x + 8 at (2, 4)
34. Find the point(s) where the graph of f(x) = x3 + x has a tangent line with slope = 0.
F.
Extrema and Critical Values (Chapter 3)
First Derivative Test: If f ’(x) > 0, the function is increasing.
If f ’(x) < 0, the function is decreasing.
If f ’(x) = 0, the function is constant.
Second Derivative Test: : If f ”(x) > 0, the function is concave up.
If f ”(x) < 0, the function is concave down.
If f ”(x) = 0, the function is linear (no concavity).
Asymptotes: a.) Vertical Asymptotes – Set denominator = 0 and solve for x.
b.) Horizontal Asymptotes – Find highest power of x. HA = y =
coefficient of highest x in num.
coefficient of highest x in den.
c.) Oblique (Slant) Asymptotes) – Long division and drop the remainder. (Only occurs if the degree of
the numerator is exactly one larger than the degree of the denominator.)
Problems:
35. Determine the intervals where f(x) = ¼ x3 – 3x is increasing.
36. Find all maxs, mins, and inflection points of f(x) = x3 – 6x2 + 15
37. Determine the intervals where f(x) = -3x3 – 9x + 1 is concave down.
38. Find all asymptotes.
2x 2
a.) f ( x) =
x2 −1
b.) f ( x ) =
x 2 − 6 x + 12
x−4
39. Find the differential, dy.
a.) y =
x −1
x+2
b.) y = x sin x
G. Integration (Chapter 4)
Rules for Integration:
n+1
x
n
1. ∫ x dx =
+C
n +1
2. ∫ kx dx = k ∫ x n dx
4. ∫ cos xdx = sin x + C
5.
n
3. ∫ sin xdx = − cos x + C
b
∫ f ( x) dx = F (b) − F (a)
a
Problems.
Evaluate each integral.
40. ∫ x dx
3 2
41. ∫ ( x + 3 x − 1) dx
2
3 3
42. ∫ x 2 dx
1
3
43. ∫ sin x cos xdx
1
3
44. ∫ ( 4 x − 2 x ) dx
−1
1 2 3
3
45. ∫ x ( x + 1) dx
0
46. ∫
3
cos x
dx
2
1 − sin x
47. ∫ 3 sin 2 xdx
H. Logarithmic and Exponential Functions (Chapter 5)
Properties of Logarithms
ln(1) = 0
log a 1 = 0
ln ab = ln a + lnb
log ab = log a + logb
a
ln = ln a − lnb
b
a
log = log a − logb
b
ln(a n ) = n ln a
log a n = n log a
Derivative of ln:
d
1
x has to be greater than 0
(ln x) =
dx
x
d
1
(lnu) = du u must be greater than 0
dx
u
Integral of ln:
1
∫ x dx = ln x + c
∫
du
1
= ∫ du = lnu + c
u
u
Derivative of e
d x
(e ) = e x
dx
d u
(e ) = eu du
dx
Integrative e
∫e
x
dx = e x + c
∫e
u
du = eu + c
Derivative for base other than e
d x
(a ) = (ln a)a x
dx
d u
du
(a ) = (ln a)a u
dx
dx
d
1
(log a x) =
dx
(ln a)x
d
1 du
(log a u) =
dx
(ln a)u dx
Integral of base other than e
⎛ ax ⎞
∫ a dx = ⎜⎝ ln a ⎟⎠ + c
x
⎛ au ⎞
∫ a du = ⎜⎝ ln a ⎟⎠ + c
u
Problems:
Find the derivative
f (x) = ln(x 2 + 2)3
1.
2.
f (x) = ln(2x + 1) + ln(x − 4)
3.
f (x) = ln
x 2 (x + 4)
(2x − 1) 2
4.
f (x) = 3x ln x
5.
f (x) = e 4 x
6.
f (x) = x 3e x
7.
f (x) = 5 3x
8.
f (x) = x 5 4 x
9.
f (x) = log 3 (x 2 − 3x)
10.
f (x) = log 4 (6x + 2)
11.
f (x) = log 2
14.
∫ 3 + 2 cos x
x4
2x + 2
12.
f (x) = x10 x
15.
∫x
Find the Integral
13.
∫e
5x
(6)dx
sin x
x+5
dx
+ 10x
2
16.
∫ 10
x 2 +5 x
(x + 2.5)dx
17.
∫e
x 2 +2 x−3
(x + 1)dx
18.
∫5
2 x 3 +3
(x 2 )dx