Download MTH 133 Final Review: Differential and Integral Calculus and more Exams Calculus in PDF only on Docsity!
MTH 133 Final Review
- Differentiation
Formulas in the first page of your text, focusing on: Chain rule, Quotient rule, Product rule, Trigonometric Functions, Exponential and Logorithmic Functions, Inverse Trigonometric Functions.
(a) y = sin−^1
1 − x^2 (b) y = (1 + t^2 ) cos−^1 (2t) (c) y = csc−^1 (sec θ) (d) y = (ln x)sin^ x (e) y = (^) (x + 2)(x^ −^ x^1 − 3)
- Integration
Formulas in Page 554 of your text. You could memorize them by combining your memory about the derivative formulas.
(a) Substation:
∫ csc^2 xecot^ xdx,
x
ln x
dx
(b) Integration by parts:
∫ ln (x + 1)dx,
∫ ex^ cos^3 2 xdx
(c) Completing the square:
x^2 − 6 x + 10 dx (d) Partial fraction:
∫ (^) x (^3) + 4x 2 x^2 + 4x + 3
dx
(e) Trigonometric substitution: involving terms like
a^2 − x^2 ,
a^2 + x^2 ,
x^2 − a^2 (f) Trigonometric integral:
∫ sin^3 x cos^4 xdx,
∫ tan^4 xdx, page 584
- Improper Integral
(a) Type I:
∫^3
0
dx x − 1
(b) Type II:
∫^ ∞
1
dx xp
(c) Direct comparison test:
∫^ ∞
1
sin^2 x x^2
dx
(d) Limit comparison test:
∫^ ∞
1
1 + x^2 dx
- Solve differential equations (separable) dy dx
= (1 + y^2 )ex^ and y(0) = 2
- Application of definite integrals, Chapter 6
(a) Volume: V =
∫^ b
a
A(x)dx
(b) Solids of revolution: V =
∫^ b
a
π(R(x))^2 )dx
(c) Washer Method: V =
∫^ b
a
π((R(x))^2 ) − (r(x))^2 )dx, page 407, 45-
(d) Length of a curve: L =
∫^ b
a
√ f ′(t)^2 + g′(t)^2 dt if x = f (t), y = g(t), and a ≤ t ≤ b;
L =
∫^ b
a
√ f ′(x)^2 + 1dx if y = f (x), and a ≤ x ≤ b; Ex. page 423,7,
(e) Work: W =
∫^ b
a
F (x), Ex. page 452-453, 1,7,15, also the example in Exam 1.
(f) Exponential growth and decay problem: y = y 0 ekt, half life, doubling time. Note: (b),(c) also need to know how to draw the graph like those you have seen in Exam 1.
- Polar Coordinates
(a) Draw the graph (b) Intersection points: set r 1 = r 2 and then solve for θ, don’t forget points intersected when r 1 and r 2 having different r and θ
(c) Area: S = (^12)
∫^ β
α
((r 2 (θ))^2 − (r 1 (θ))^2 )dθ. clue: when forming the area, how θ changes?
Check examples in sample final and exam 3.
