topics: trig substitution: substitute and integrate...

Topics: Trig Substitution: Substitute and Integrate Partial Fractions: Decompose and Integrate Numerical Integration: Trapezoid, Midpoint, Simpson Know the formulas. Error Polar: Convert between rectangular and polar (equations and points) Area Sketching: flowers, lines, cardiods, lemiçons, circles Intersections of graphs Parametrics: Eliminate the parameter, slopes, vertical/horizontal tangents, tangent lines

Sequences: Limits, boundedness, increasing, decreasing, GLB, LUB

Section 9.3 Polar Coordinates Give 4 possible polar coordinates for the given point. Give ALL possible polar coordinates for the given point. 1. 2. (–4, 4)

( )3 1,−

Give rectangular coordinates for:

x = r cos θ y = r sin θ 1. (2, π) 2.

36, π⎛ ⎞

⎜ ⎟⎝ ⎠

Give polar coordinates for: 1. (0, 2) 2. (–3, –3)

Write the equation in polar coordinates. r2 = x2 + y2 tan θ = y/x 1. x2 + y2 = 16 2. y = x 3. x = 10 4. x2 + (y + 3)2 = 9

Write in rectangular coordinates. 1. r = 3 sinθ 2. tanθ = 5 3. r sinθ = 6

9.4 Graphing in Polar Coordinates Sketch the polar curve for the given values of �. 1. r = 3 + 3 cosθ 2. r = 2 – 2 cosθ

0 ≤ θ ≤ π

32 2π π≤ θ ≤

3. r = 3 + 1 sinθ 4. r = 1 – 3 sinθ

2π ≤ θ ≤ π

2π ≤ θ ≤ π

5. r = cos (4θ) 6. r = sin (3θ)

02π≤ θ ≤

2π ≤ θ ≤ π

Section 9.5 Area in Polar Coordinates Set up the integral that gives the requested area. 1. one petal of r = 3 cos (5θ) 2. one petal of r = 2 cos (4θ)

3. inside outer loop of r = 1 – 2 cos θ and in Quadrants I, IV 4. one petal of r = 5 sin (3θ)

5. outside r = 2 and inside r = 4 sin θ 6. inside r = 2 and outside r = 4 cos θ

7. inside the inner loop of r = 1 – 2 sin θ

Section 9.6 Curves given Parametrically Give an equation relating x and y. 1. x(t) = sin t y(t) = 1 + cos t 2. x(t) = 3 + 2cos t y(t) = –1 + sin t

1. Give a parameterization for the line segment from (–5, 8) to (1, 3).

Section 9.7 Tangents to curves given parametrically Find an equation in x and y for the line tangent (or normal) to the curve. 1. x(t) = 2 – 3 cos t y(t) = 3 + 2 sin t at t = π/4

dydy dt

dxdxdt

=

2. Find points of horizontal and vertical tangency. x(t) = t2 y(t) = t3 – 3t

Section 9.7 Tangents to curves given parametrically Find an equation in x and y for the line tangent (or normal) to the curve. 1. x(t) = 2 – 3 cos t y(t) = 3 + 2 sin t at t = π/4

dydy dt

dxdxdt

=

2. Find points of horizontal and vertical tangency. x(t) = t2 y(t) = t3 – 3t

3. Find a parametrization for the curve y3 = x2 from (1, 1) to (8, 4).

4. Give an equation for the normal line to the graph of x = sin t, y = 2 + cos 2t at the point where t = π/6.

5. Give an equation for the line tangent to the polar curve r = 2 cos θ at the point where θ = π/3.

POPPER

1. Give the limit for ( )cos

n 1

n

n

∞

=

⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭

a. 0 b. 1 c. 2 d. 3 e. none of these

POPPER 2. Give the y intercept of the normal line to (t2 – 1, t + t2) at (0, 2). a. 0 b. 1 c. 2 d. 3 e. none of these

POPPER 3. Give a parameterization for the line segment from (2, 1) to (–4, 3). a. x(t) = 2 – 3t, y(t) = 1 + t, 0 ≤ t ≤ 1 b. x(t) = –4+6t, y(t) = 3 – 2t, 0 ≤ t ≤ 1 c. x(t) = 2 – 6t, y(t) = 1 + 2t, 0 ≤ t ≤ 1 d. x(t) = – 4 + 3t, y(t) = 3 – t, 0 ≤ t ≤ 1

POPPER

4. Give the GLB of n

n 1

12

∞

=

⎧ ⎫−⎛ ⎞⎪ ⎪⎨ ⎬⎜ ⎟⎝ ⎠⎪ ⎪⎩ ⎭

a. –1 b. 0 c. 1 d. DNE e. none of these

POPPER

5. Give the limit for 2

n 1

nn 1

∞

=

⎧ ⎫⎪ ⎪⎨ ⎬+⎪ ⎪⎩ ⎭

a. 0 b. 1 c. 2 d. DNE e. none of these

POPPER

6. Give the limit for 1

4n

−

a. 1 b. 2 c. 3 d. DNE e. none of these

POPPER

7. Give the limit for tan n4n 1

π+

a. 0 b. 1 c. 2 d. DNE e. none of these

POPPER

8. Give the limit for n

n

2

4 1+

a. 0 b. 1 c. ½ d. DNE e. none of these

POPPER

9. Give the limit for 2n

11n

⎛ ⎞+⎜ ⎟⎝ ⎠

a. 1 b. e c. e2 d. DNE e. none of these