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AP Unit 9 Problems

Thirty problems for AP Unit 9. Try the problem before opening its separate solution. Write the mathematical setup and the reason your method applies.

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Topic Learning objective Calculator practice Representation

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Problem 1

Topic 9.1 · No calculator · analytic

For $x = t^{2} + 1$ , $y = t^{3} - 3 t$ , find the tangent line at $t = 2$ .

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Problem 2

Topic 9.1 · No calculator · analytic

For $x = t^{2}$ , $y = t$ , classify the tangent at $t = 0$ and explain its equation.

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Problem 3

Topic 9.1 · No calculator · multiple

The graph is the ellipse $x = 2 \cos t$ , $y = 3 \sin t$ , traced counterclockwise from (2,0). At $t = π / 2$ , find the point, tangent and instantaneous direction of motion.

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Problem 4

Topic 9.2 · No calculator · analytic

For $x = t^{2}$ , $y = t^{3}$ , find $d^{2} y / d x^{2}$ at $t = 2$ .

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Problem 5

Topic 9.2 · No calculator · analytic

For $x = t^{3} - 3 t$ , $y = t^{2}$ , determine concavity at $t = 0$ .

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Problem 6

Topic 9.2 · No calculator · verbal

A curve has $x = t^{2}$ , $y = t^{3}$ on $t < 0$ . Is its y-versus-x graph concave up or down? State the domain restriction behind your calculation.

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Problem 7

Topic 9.3 · No calculator · analytic

Find the length of $x = 3 t$ , $y = 4 t$ for $0 \leq t \leq 2$ .

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Problem 8

Topic 9.3 · No calculator · multiple

Find the arc length of $x = \cos t$ , $y = \sin t$ on $0 \leq t \leq π$ . Describe which part of the curve is traced.

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Problem 9

Topic 9.3 9.6 · No calculator · verbal

The unit circle $〈 \cos t, \sin t 〉$ is traced on $0 \leq t \leq 4 π$ . Find distance traveled and the length of the distinct geometric curve. Explain the difference.

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Problem 10

Topic 9.4 · No calculator · analytic

Given $r (t) = 〈 t^{2}, e^{t} 〉$ , find velocity, acceleration and speed at $t = 0$ .

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Problem 11

Topic 9.4 · No calculator · analytic

For $r (t) = 〈 \sin (2 t), \cos (2 t) 〉$ , determine speed and acceleration magnitude at any t.

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Problem 12

Topic 9.4 9.1 · No calculator · numerical

A derivative table gives instantaneous rates at time 2.

Exact coordinate derivative values at t=2
t	x′(t)	y′(t)
2	3	−4

Find the velocity, speed and tangent slope. Can this table determine the particle's position?

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Problem 13

Topic 9.5 · No calculator · analytic

Given $v (t) = 〈 2 t, 3 〉$ and $r (1) = 〈 4, 5 〉$ , find r(t) and r(3).

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Problem 14

Topic 9.5 · No calculator · analytic

A particle has $a (t) = 〈 6 t, - 2 〉$ , $v (0) = 〈 1, 4 〉$ , and $r (0) = 〈 2, - 1 〉$ . Find its position.

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Problem 15

Topic 9.5 9.6 · No calculator · numerical

A particle starts at $〈 1, 4 〉$ at t=0. Use the trapezoidal rule on each component to estimate its position at t=2.

Velocity samples; meters per second
t (s)	x′	y′
0	0	2
1	2	0
2	2	−2

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Problem 16

Topic 9.6 · No calculator · analytic

For $r (t) = 〈 t^{2}, t^{3} 〉$ , find speed at t=1 and decide whether the particle is speeding up there.

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Problem 17

Topic 9.6 · No calculator · verbal

A particle follows $r (t) = 〈 \cos t, \sin t 〉$ from t=0 to t=2π. Find displacement and total distance, then explain why they differ.

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Problem 18

Topic 9.6 · No calculator · verbal

A particle moves with constant velocity $〈 3, 4 〉$ m/s for 2 seconds. Find displacement, distance and average velocity.

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Problem 19

Topic 9.7 · No calculator · analytic

On $r = 2$ , find the Cartesian point and tangent slope at $θ = π / 3$ . Explain why dr/dθ=0 does not give the slope.

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Problem 20

Topic 9.7 9.2 · No calculator · analytic

For $r = 1 + \cos θ$ , compute $d y / d x$ and $d^{2} y / d x^{2}$ at $θ = π / 2$ .

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Problem 21

Topic 9.7 · No calculator · analytic

On $r = 2 \sin θ$ , classify the tangent at $θ = π / 4$ and give its line.

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Problem 22

Topic 9.7 · No calculator · graphical

A polar graph marks radius −2 at angle 0 on the curve $r = - 2$ . Locate the point in Cartesian coordinates, give an equivalent polar pair with positive radius, and classify the tangent there.

The signed radius points opposite the chosen ray.

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Problem 23

Topic 9.8 · No calculator · analytic

Find the area of the full disk bounded by $r = 3$ .

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Problem 24

Topic 9.8 · No calculator · multiple

The curve $r = 2 \cos θ$ is a unit circle centered at (1,0). Set bounds that trace it once and calculate its enclosed area.

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Problem 25

Topic 9.8 · No calculator · analytic

Find the area of one petal of $r = 2 \sin (3 θ)$ swept from θ=0 to θ=π/3. Explain why these bounds select one petal.

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Problem 26

Topic 9.8 · No calculator · analytic

Find the total enclosed area of the cardioid $r = 1 + \cos θ$ , traced once on $0 \leq θ \leq 2 π$ .

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Problem 27

Topic 9.9 · No calculator · analytic

Find the area inside $r = 2$ and outside $r = 1 + \cos θ$ . State why one difference integral works on the full turn.

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Problem 28

Topic 9.9 · No calculator · analytic

Find the area inside $r = 2 \sin θ$ but outside $r = 1$ in the upper half-plane.

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Problem 29

Topic 9.9 · No calculator · multiple

Find the overlap area inside both $r = 1$ and $r = 2 \cos θ$ . Your solution must explain a change of boundary.

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Problem 30

Topic 9.9 · Graphing calculator required · analytic

Use numerical integration to find, to three decimals, the area in $0 \leq θ \leq π$ inside $r = 2 + \sin θ$ and outside $r = 1$ . Give the defining integral and justify the order of the boundaries. Use radians.

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