Topics: Trig Substitution: Substitute and Integrate...
Transcript of Topics: Trig Substitution: Substitute and Integrate...
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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
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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,−
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Give rectangular coordinates for:
x = r cos θ y = r sin θ 1. (2, π) 2.
36, π⎛ ⎞
⎜ ⎟⎝ ⎠
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Give polar coordinates for: 1. (0, 2) 2. (–3, –3)
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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
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Write in rectangular coordinates. 1. r = 3 sinθ 2. tanθ = 5 3. r sinθ = 6
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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 ≤
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3. r = 3 + 1 sinθ 4. r = 1 – 3 sinθ
2π ≤ θ ≤ π
2π ≤ θ ≤ π
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5. r = cos (4θ) 6. r = sin (3θ)
02π≤ θ ≤
2π ≤ θ ≤ π
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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)
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3. inside outer loop of r = 1 – 2 cos θ and in Quadrants I, IV 4. one petal of r = 5 sin (3θ)
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5. outside r = 2 and inside r = 4 sin θ 6. inside r = 2 and outside r = 4 cos θ
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7. inside the inner loop of r = 1 – 2 sin θ
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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
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1. Give a parameterization for the line segment from (–5, 8) to (1, 3).
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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
=
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2. Find points of horizontal and vertical tangency. x(t) = t2 y(t) = t3 – 3t
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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
=
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2. Find points of horizontal and vertical tangency. x(t) = t2 y(t) = t3 – 3t
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3. Find a parametrization for the curve y3 = x2 from (1, 1) to (8, 4).
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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.
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5. Give an equation for the line tangent to the polar curve r = 2 cos θ at the point where θ = π/3.
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POPPER
1. Give the limit for ( )cos
n 1
n
n
∞
=
⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭
a. 0 b. 1 c. 2 d. 3 e. none of these
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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
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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
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POPPER
4. Give the GLB of n
n 1
12
∞
=
⎧ ⎫−⎛ ⎞⎪ ⎪⎨ ⎬⎜ ⎟⎝ ⎠⎪ ⎪⎩ ⎭
a. –1 b. 0 c. 1 d. DNE e. none of these
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POPPER
5. Give the limit for 2
n 1
nn 1
∞
=
⎧ ⎫⎪ ⎪⎨ ⎬+⎪ ⎪⎩ ⎭
a. 0 b. 1 c. 2 d. DNE e. none of these
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POPPER
6. Give the limit for 1
4n
−
a. 1 b. 2 c. 3 d. DNE e. none of these
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POPPER
7. Give the limit for tan n4n 1
π+
a. 0 b. 1 c. 2 d. DNE e. none of these
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POPPER
8. Give the limit for n
n
2
4 1+
a. 0 b. 1 c. ½ d. DNE e. none of these
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POPPER
9. Give the limit for 2n
11n
⎛ ⎞+⎜ ⎟⎝ ⎠
a. 1 b. e c. e2 d. DNE e. none of these