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Horizontal and vertical tangents of parametric curves practice problems

Points where a parametric curve has a horizontal or a vertical tangent. 62 problems with worked solutions; in 62 of them every equation is proved by a computer algebra system.

7 worked problems

Try each one before opening the solution. Every line is labeled with what checked it. How we verify

Problem 8.241

medium
Find the points where \( \displaystyle x = t^{3} - 6 t,\ y = t^{3} - 3 t \) has a horizontal or a vertical tangent line.
  1. \[ \frac{d}{d t} \left(t^{3} - 6 t\right) = 3 t^{2} - 6 \]
    dx/dt.✓ Proved
  2. \[ \frac{d}{d t} \left(t^{3} - 3 t\right) = 3 t^{2} - 3 \]
    dy/dt.✓ Proved
  3. Horizontal tangent: dy/dt = 0 with dx/dt ≠ 0. Vertical: dx/dt = 0 with dy/dt ≠ 0.
    Reviewed
  4. \[ \left. 3 t^{2} - 3 \right|_{\substack{ t=-1 }} = 0 \]
    t = -1: horizontal, at (5, 2).✓ Proved
  5. \[ \left. 3 t^{2} - 3 \right|_{\substack{ t=1 }} = 0 \]
    t = 1: horizontal, at (-5, -2).✓ Proved
  6. \[ \left. 3 t^{2} - 6 \right|_{\substack{ t=- \sqrt{2} }} = 0 \]
    t = -sqrt(2): vertical, at (4*sqrt(2), sqrt(2)).✓ Proved
  7. \[ \left. 3 t^{2} - 6 \right|_{\substack{ t=\sqrt{2} }} = 0 \]
    t = sqrt(2): vertical, at (-4*sqrt(2), -sqrt(2)).✓ Proved
Answer \( \text{horizontal at } (5, 2), (-5, -2);\ \text{vertical at } (4*sqrt(2), sqrt(2)), (-4*sqrt(2), -sqrt(2)) \)

✓ Nihil obstat Lines: 6 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 8.235

medium
Find the points where \( \displaystyle x = t^{3} - 6 t,\ y = t^{2} + 1 \) has a horizontal or a vertical tangent line.
  1. \[ \frac{d}{d t} \left(t^{3} - 6 t\right) = 3 t^{2} - 6 \]
    dx/dt.✓ Proved
  2. \[ \frac{d}{d t} \left(t^{2} + 1\right) = 2 t \]
    dy/dt.✓ Proved
  3. Horizontal tangent: dy/dt = 0 with dx/dt ≠ 0. Vertical: dx/dt = 0 with dy/dt ≠ 0.
    Reviewed
  4. \[ \left. 2 t \right|_{\substack{ t=0 }} = 0 \]
    t = 0: horizontal, at (0, 1).✓ Proved
  5. \[ \left. 3 t^{2} - 6 \right|_{\substack{ t=- \sqrt{2} }} = 0 \]
    t = -sqrt(2): vertical, at (4*sqrt(2), 3).✓ Proved
  6. \[ \left. 3 t^{2} - 6 \right|_{\substack{ t=\sqrt{2} }} = 0 \]
    t = sqrt(2): vertical, at (-4*sqrt(2), 3).✓ Proved
Answer \( \text{horizontal at } (0, 1);\ \text{vertical at } (4*sqrt(2), 3), (-4*sqrt(2), 3) \)

✓ Nihil obstat Lines: 5 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 8.249

medium
Find the points where \( \displaystyle x = t^{3} - 3 t,\ y = t^{2} - 4 t + 2 \) has a horizontal or a vertical tangent line.
  1. \[ \frac{d}{d t} \left(t^{3} - 3 t\right) = 3 t^{2} - 3 \]
    dx/dt.✓ Proved
  2. \[ \frac{d}{d t} \left(t^{2} - 4 t + 2\right) = 2 t - 4 \]
    dy/dt.✓ Proved
  3. Horizontal tangent: dy/dt = 0 with dx/dt ≠ 0. Vertical: dx/dt = 0 with dy/dt ≠ 0.
    Reviewed
  4. \[ \left. 2 t - 4 \right|_{\substack{ t=2 }} = 0 \]
    t = 2: horizontal, at (2, -2).✓ Proved
  5. \[ \left. 3 t^{2} - 3 \right|_{\substack{ t=-1 }} = 0 \]
    t = -1: vertical, at (2, 7).✓ Proved
  6. \[ \left. 3 t^{2} - 3 \right|_{\substack{ t=1 }} = 0 \]
    t = 1: vertical, at (-2, -1).✓ Proved
Answer \( \text{horizontal at } (2, -2);\ \text{vertical at } (2, 7), (-2, -1) \)

✓ Nihil obstat Lines: 5 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 8.374

medium
Find the points where \( \displaystyle x = t^{3} - 12 t,\ y = 3 t^{2} - 12 \) has a horizontal or a vertical tangent line.
  1. \[ \frac{d}{d t} \left(t^{3} - 12 t\right) = 3 t^{2} - 12 \]
    dx/dt.✓ Proved
  2. \[ \frac{d}{d t} \left(3 t^{2} - 12\right) = 6 t \]
    dy/dt.✓ Proved
  3. Horizontal tangent: dy/dt = 0 with dx/dt ≠ 0. Vertical: dx/dt = 0 with dy/dt ≠ 0.
    Reviewed
  4. \[ \left. 6 t \right|_{\substack{ t=0 }} = 0 \]
    t = 0: horizontal, at (0, -12).✓ Proved
  5. \[ \left. 3 t^{2} - 12 \right|_{\substack{ t=-2 }} = 0 \]
    t = -2: vertical, at (16, 0).✓ Proved
  6. \[ \left. 3 t^{2} - 12 \right|_{\substack{ t=2 }} = 0 \]
    t = 2: vertical, at (-16, 0).✓ Proved
Answer \( \text{horizontal at } (0, -12);\ \text{vertical at } (16, 0), (-16, 0) \)

✓ Nihil obstat Lines: 5 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 8.376

medium
Find the points where \( \displaystyle x = t^{3} - 3 t,\ y = t^{2} - 4 t - 3 \) has a horizontal or a vertical tangent line.
  1. \[ \frac{d}{d t} \left(t^{3} - 3 t\right) = 3 t^{2} - 3 \]
    dx/dt.✓ Proved
  2. \[ \frac{d}{d t} \left(t^{2} - 4 t - 3\right) = 2 t - 4 \]
    dy/dt.✓ Proved
  3. Horizontal tangent: dy/dt = 0 with dx/dt ≠ 0. Vertical: dx/dt = 0 with dy/dt ≠ 0.
    Reviewed
  4. \[ \left. 2 t - 4 \right|_{\substack{ t=2 }} = 0 \]
    t = 2: horizontal, at (2, -7).✓ Proved
  5. \[ \left. 3 t^{2} - 3 \right|_{\substack{ t=-1 }} = 0 \]
    t = -1: vertical, at (2, 2).✓ Proved
  6. \[ \left. 3 t^{2} - 3 \right|_{\substack{ t=1 }} = 0 \]
    t = 1: vertical, at (-2, -6).✓ Proved
Answer \( \text{horizontal at } (2, -7);\ \text{vertical at } (2, 2), (-2, -6) \)

✓ Nihil obstat Lines: 5 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 8.381

medium
Find the points where \( \displaystyle x = t^{3} - 3 t,\ y = t^{2} - 2 \) has a horizontal or a vertical tangent line.
  1. \[ \frac{d}{d t} \left(t^{3} - 3 t\right) = 3 t^{2} - 3 \]
    dx/dt.✓ Proved
  2. \[ \frac{d}{d t} \left(t^{2} - 2\right) = 2 t \]
    dy/dt.✓ Proved
  3. Horizontal tangent: dy/dt = 0 with dx/dt ≠ 0. Vertical: dx/dt = 0 with dy/dt ≠ 0.
    Reviewed
  4. \[ \left. 2 t \right|_{\substack{ t=0 }} = 0 \]
    t = 0: horizontal, at (0, -2).✓ Proved
  5. \[ \left. 3 t^{2} - 3 \right|_{\substack{ t=-1 }} = 0 \]
    t = -1: vertical, at (2, -1).✓ Proved
  6. \[ \left. 3 t^{2} - 3 \right|_{\substack{ t=1 }} = 0 \]
    t = 1: vertical, at (-2, -1).✓ Proved
Answer \( \text{horizontal at } (0, -2);\ \text{vertical at } (2, -1), (-2, -1) \)

✓ Nihil obstat Lines: 5 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 8.582

medium
Find the points where \( \displaystyle x = t^{3} - 6 t,\ y = t^{2} \) has a horizontal or a vertical tangent line.
  1. \[ \frac{d}{d t} \left(t^{3} - 6 t\right) = 3 t^{2} - 6 \]
    dx/dt.✓ Proved
  2. \[ \frac{d}{d t} t^{2} = 2 t \]
    dy/dt.✓ Proved
  3. Horizontal tangent: dy/dt = 0 with dx/dt ≠ 0. Vertical: dx/dt = 0 with dy/dt ≠ 0.
    Reviewed
  4. \[ \left. 2 t \right|_{\substack{ t=0 }} = 0 \]
    t = 0: horizontal, at (0, 0).✓ Proved
  5. \[ \left. 3 t^{2} - 6 \right|_{\substack{ t=- \sqrt{2} }} = 0 \]
    t = -sqrt(2): vertical, at (4*sqrt(2), 2).✓ Proved
  6. \[ \left. 3 t^{2} - 6 \right|_{\substack{ t=\sqrt{2} }} = 0 \]
    t = sqrt(2): vertical, at (-4*sqrt(2), 2).✓ Proved
Answer \( \text{horizontal at } (0, 0);\ \text{vertical at } (4*sqrt(2), 2), (-4*sqrt(2), 2) \)

✓ Nihil obstat Lines: 5 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

More horizontal and vertical tangents of parametric curves practice

Find the points where \( \displaystyle x = t^{3} - 3 t,\ y = t^{2} - 3 \) has a horizontal or a vertical tangent line.
Problem 8.236medium✓ Every equation proved
Find the points where \( \displaystyle x = t^{3} - 3 t,\ y = t^{2} - 1 \) has a horizontal or a vertical tangent line.
Problem 8.237medium✓ Every equation proved
Find the points where \( \displaystyle x = t^{3} - 9 t,\ y = 3 t^{2} - 9 \) has a horizontal or a vertical tangent line.
Problem 8.238medium✓ Every equation proved
Find the points where \( \displaystyle x = t^{2} - 2 t,\ y = 3 t^{2} - 9 \) has a horizontal or a vertical tangent line.
Problem 8.239medium✓ Nihil obstat
Find the points where \( \displaystyle x = t^{2} - 2 t,\ y = 3 t^{2} - 6 \) has a horizontal or a vertical tangent line.
Problem 8.240medium✓ Nihil obstat
Find the points where \( \displaystyle x = t^{3} - 3 t,\ y = 3 t^{2} - 3 \) has a horizontal or a vertical tangent line.
Problem 8.242medium✓ Every equation proved
Find the points where \( \displaystyle x = t^{3} - 6 t,\ y = 3 t^{2} - 6 \) has a horizontal or a vertical tangent line.
Problem 8.243medium✓ Every equation proved
Find the points where \( \displaystyle x = t^{3} - 3 t,\ y = 3 t^{2} - 12 \) has a horizontal or a vertical tangent line.
Problem 8.244medium✓ Every equation proved
Find the points where \( \displaystyle x = t^{2} - 2 t,\ y = 3 t^{2} - 12 \) has a horizontal or a vertical tangent line.
Problem 8.245medium✓ Nihil obstat
Find the points where \( \displaystyle x = t^{2} - 2 t,\ y = t^{2} + 2 \) has a horizontal or a vertical tangent line.
Problem 8.246medium✓ Nihil obstat
Find the points where \( \displaystyle x = t^{3} - 3 t,\ y = t^{2} - 4 t \) has a horizontal or a vertical tangent line.
Problem 8.247medium✓ Every equation proved
Find the points where \( \displaystyle x = 3 \cos{\left(t \right)},\ y = 3 \sin{\left(2 t \right)} \) for \( \displaystyle 0 \le t < 2\pi \) has a horizontal or a vertical tangent line.
Problem 8.248medium✓ Every equation proved
Find the points where \( \displaystyle x = t^{2} - 2 t,\ y = t^{3} - 3 t \) has a horizontal or a vertical tangent line.
Problem 8.348medium✓ Every equation proved
Find the points where \( \displaystyle x = t^{3} - 9 t,\ y = t^{3} - 3 t \) has a horizontal or a vertical tangent line.
Problem 8.349medium✓ Nihil obstat
Find the points where \( \displaystyle x = t^{2} - 2 t,\ y = t^{2} + 1 \) has a horizontal or a vertical tangent line.
Problem 8.350medium✓ Nihil obstat
Find the points where \( \displaystyle x = t^{3} - 12 t,\ y = t^{3} - 3 t \) has a horizontal or a vertical tangent line.
Problem 8.351medium✓ Nihil obstat
Find the points where \( \displaystyle x = t^{2} - 2 t,\ y = t^{2} - 4 t - 2 \) has a horizontal or a vertical tangent line.
Problem 8.352medium✓ Nihil obstat
Find the points where \( \displaystyle x = t^{2} - 2 t,\ y = 3 t^{2} - 3 \) has a horizontal or a vertical tangent line.
Problem 8.353medium✓ Every equation proved
Find the points where \( \displaystyle x = 2 \cos{\left(t \right)},\ y = 2 \sin{\left(2 t \right)} \) for \( \displaystyle 0 \le t < 2\pi \) has a horizontal or a vertical tangent line.
Problem 8.354medium✓ Every equation proved
Find the points where \( \displaystyle x = t^{3} - 3 t,\ y = t^{2} - 4 t - 1 \) has a horizontal or a vertical tangent line.
Problem 8.355medium✓ Every equation proved
Find the points where \( \displaystyle x = \cos{\left(t \right)},\ y = \sin{\left(2 t \right)} \) for \( \displaystyle 0 \le t < 2\pi \) has a horizontal or a vertical tangent line.
Problem 8.356medium✓ Every equation proved
Find the points where \( \displaystyle x = t^{2} - 2 t,\ y = t^{2} - 4 t - 1 \) has a horizontal or a vertical tangent line.
Problem 8.357medium✓ Nihil obstat
Find the points where \( \displaystyle x = 4 \cos{\left(t \right)},\ y = 4 \sin{\left(2 t \right)} \) for \( \displaystyle 0 \le t < 2\pi \) has a horizontal or a vertical tangent line.
Problem 8.373medium✓ Every equation proved
Find the points where \( \displaystyle x = t^{3} - 9 t,\ y = t^{2} - 4 t + 2 \) has a horizontal or a vertical tangent line.
Problem 8.375medium✓ Nihil obstat
Find the points where \( \displaystyle x = t^{3} - 3 t,\ y = 3 t^{2} - 9 \) has a horizontal or a vertical tangent line.
Problem 8.377medium✓ Every equation proved
Find the points where \( \displaystyle x = t^{3} - 3 t,\ y = t^{2} - 4 t + 3 \) has a horizontal or a vertical tangent line.
Problem 8.378medium✓ Nihil obstat
Find the points where \( \displaystyle x = t^{3} - 3 t,\ y = t^{2} + 1 \) has a horizontal or a vertical tangent line.
Problem 8.379medium✓ Every equation proved
Find the points where \( \displaystyle x = t^{3} - 6 t,\ y = t^{2} + 2 \) has a horizontal or a vertical tangent line.
Problem 8.380medium✓ Every equation proved
Find the points where \( \displaystyle x = t^{3} - 6 t,\ y = t^{2} - 3 \) has a horizontal or a vertical tangent line.
Problem 8.382medium✓ Every equation proved
Find the points where \( \displaystyle x = t^{2} - 2 t,\ y = t^{2} - 4 t + 2 \) has a horizontal or a vertical tangent line.
Problem 8.383medium✓ Nihil obstat
Find the points where \( \displaystyle x = t^{3} - 3 t,\ y = t^{2} + 3 \) has a horizontal or a vertical tangent line.
Problem 8.384medium✓ Every equation proved
Find the points where \( \displaystyle x = t^{3} - 9 t,\ y = t^{2} + 2 \) has a horizontal or a vertical tangent line.
Problem 8.385medium✓ Nihil obstat
Find the points where \( \displaystyle x = t^{3} - 3 t,\ y = t^{2} - 4 t - 2 \) has a horizontal or a vertical tangent line.
Problem 8.386medium✓ Nihil obstat
Find the points where \( \displaystyle x = t^{2} - 2 t,\ y = t^{2} - 4 t + 1 \) has a horizontal or a vertical tangent line.
Problem 8.387medium✓ Nihil obstat
Find the points where \( \displaystyle x = t^{3} - 9 t,\ y = t^{2} - 2 \) has a horizontal or a vertical tangent line.
Problem 8.495medium✓ Nihil obstat
Find the points where \( \displaystyle x = t^{2} - 2 t,\ y = t^{2} - 2 \) has a horizontal or a vertical tangent line.
Problem 8.496medium✓ Nihil obstat
Find the points where \( \displaystyle x = t^{2} - 2 t,\ y = t^{2} + 3 \) has a horizontal or a vertical tangent line.
Problem 8.497medium✓ Nihil obstat
Find the points where \( \displaystyle x = t^{3} - 3 t,\ y = t^{2} \) has a horizontal or a vertical tangent line.
Problem 8.498medium✓ Every equation proved
Find the points where \( \displaystyle x = t^{2} - 2 t,\ y = t^{2} - 1 \) has a horizontal or a vertical tangent line.
Problem 8.499medium✓ Nihil obstat
Find the points where \( \displaystyle x = t^{2} - 2 t,\ y = t^{2} - 3 \) has a horizontal or a vertical tangent line.
Problem 8.500medium✓ Nihil obstat
Find the points where \( \displaystyle x = t^{3} - 3 t,\ y = t^{2} + 2 \) has a horizontal or a vertical tangent line.
Problem 8.501medium✓ Every equation proved
Find the points where \( \displaystyle x = t^{3} - 9 t,\ y = t^{2} - 4 t - 2 \) has a horizontal or a vertical tangent line.
Problem 8.502medium✓ Every equation proved
Find the points where \( \displaystyle x = t^{2} - 2 t,\ y = t^{2} - 4 t - 3 \) has a horizontal or a vertical tangent line.
Problem 8.503medium✓ Nihil obstat
Find the points where \( \displaystyle x = t^{3} - 9 t,\ y = t^{2} - 1 \) has a horizontal or a vertical tangent line.
Problem 8.504medium✓ Nihil obstat
Find the points where \( \displaystyle x = t^{3} - 12 t,\ y = t^{2} - 4 t - 1 \) has a horizontal or a vertical tangent line.
Problem 8.541medium✓ Every equation proved
Find the points where \( \displaystyle x = t^{2} - 2 t,\ y = t^{2} - 4 t + 3 \) has a horizontal or a vertical tangent line.
Problem 8.542medium✓ Nihil obstat
Find the points where \( \displaystyle x = t^{3} - 6 t,\ y = t^{2} + 3 \) has a horizontal or a vertical tangent line.
Problem 8.573medium✓ Nihil obstat
Find the points where \( \displaystyle x = t^{2} - 2 t,\ y = t^{2} \) has a horizontal or a vertical tangent line.
Problem 8.574medium✓ Nihil obstat
Find the points where \( \displaystyle x = t^{3} - 12 t,\ y = t^{2} + 1 \) has a horizontal or a vertical tangent line.
Problem 8.575medium✓ Nihil obstat
Find the points where \( \displaystyle x = t^{3} - 6 t,\ y = t^{2} - 1 \) has a horizontal or a vertical tangent line.
Problem 8.576medium✓ Every equation proved
Find the points where \( \displaystyle x = t^{3} - 12 t,\ y = t^{2} - 4 t + 3 \) has a horizontal or a vertical tangent line.
Problem 8.577medium✓ Every equation proved
Find the points where \( \displaystyle x = t^{3} - 12 t,\ y = t^{2} - 4 t + 1 \) has a horizontal or a vertical tangent line.
Problem 8.578medium✓ Every equation proved
Find the points where \( \displaystyle x = t^{3} - 3 t,\ y = t^{2} - 4 t + 1 \) has a horizontal or a vertical tangent line.
Problem 8.579medium✓ Nihil obstat
Find the points where \( \displaystyle x = t^{3} - 6 t,\ y = t^{2} - 4 t + 3 \) has a horizontal or a vertical tangent line.
Problem 8.580medium✓ Nihil obstat
Find the points where \( \displaystyle x = t^{3} - 9 t,\ y = t^{2} + 3 \) has a horizontal or a vertical tangent line.
Problem 8.581medium✓ Nihil obstat