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Tangent lines to parametric curves practice problems

dy/dx = (dy/dt)/(dx/dt), and the tangent line at a parameter value. 70 problems with worked solutions; in 70 of them every equation is proved by a computer algebra system.

20 worked problems

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

Problem 8.1

hard
Find the equation of the tangent line to \( \displaystyle x = e^{t},\ y = t^{3} \) at \( \displaystyle t = 1 \).
  1. dy/dx = (dy/dt)/(dx/dt).
    Reviewed
  2. \[ \frac{d}{d t} e^{t} = e^{t} \]
    dx/dt.✓ Proved
  3. \[ \frac{d}{d t} t^{3} = 3 t^{2} \]
    dy/dt.✓ Proved
  4. \[ \frac{3}{e} \]
    The slope at t = 1.Reviewed
  5. \[ \left[\begin{matrix}e\\1\end{matrix}\right] \]
    The point.Reviewed
  6. \[ \frac{3 x - 3 e}{e} + 1 = \frac{3 x}{e} - 2 \]
    Point-slope form.✓ Proved
Answer \( y = \frac{3 x}{e} - 2 \)

✓ Nihil obstat Lines: 3 proved, 3 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 8.2

hard
Find the equation of the tangent line to \( \displaystyle x = t^{3} - t,\ y = t^{3} - 3 t \) at \( \displaystyle t = 2 \).
  1. dy/dx = (dy/dt)/(dx/dt).
    Reviewed
  2. \[ \frac{d}{d t} \left(t^{3} - t\right) = 3 t^{2} - 1 \]
    dx/dt.✓ Proved
  3. \[ \frac{d}{d t} \left(t^{3} - 3 t\right) = 3 t^{2} - 3 \]
    dy/dt.✓ Proved
  4. \[ \frac{9}{11} \]
    The slope at t = 2.Reviewed
  5. \[ \left[\begin{matrix}6\\2\end{matrix}\right] \]
    The point.Reviewed
  6. \[ \frac{9 x}{11} - \frac{32}{11} \]
    Point-slope form.Reviewed
Answer \( y = \frac{9 x}{11} - \frac{32}{11} \)

✓ Nihil obstat Lines: 2 proved, 4 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 8.3

hard
Find the equation of the tangent line to \( \displaystyle x = 2 t,\ y = t^{2} - 3 t \) at \( \displaystyle t = 1 \).
  1. dy/dx = (dy/dt)/(dx/dt).
    Reviewed
  2. \[ \frac{d}{d t} 2 t = 2 \]
    dx/dt.✓ Proved
  3. \[ \frac{d}{d t} \left(t^{2} - 3 t\right) = 2 t - 3 \]
    dy/dt.✓ Proved
  4. \[ - \frac{1}{2} \]
    The slope at t = 1.Reviewed
  5. \[ \left[\begin{matrix}2\\-2\end{matrix}\right] \]
    The point.Reviewed
  6. \[ \frac{\left(-1\right) x}{2} - 1 = - \frac{x}{2} - 1 \]
    Point-slope form.Reviewed
Answer \( y = - \frac{x}{2} - 1 \)

✓ Nihil obstat Lines: 2 proved, 4 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 8.4

hard
Find the equation of the tangent line to \( \displaystyle x = e^{t},\ y = 3 t + 1 \) at \( \displaystyle t = 1 \).
  1. dy/dx = (dy/dt)/(dx/dt).
    Reviewed
  2. \[ \frac{d}{d t} e^{t} = e^{t} \]
    dx/dt.✓ Proved
  3. \[ \frac{d}{d t} \left(3 t + 1\right) = 3 \]
    dy/dt.✓ Proved
  4. \[ \frac{3}{e} \]
    The slope at t = 1.Reviewed
  5. \[ \left[\begin{matrix}e\\4\end{matrix}\right] \]
    The point.Reviewed
  6. \[ \frac{3 x - 3 e}{e} + 4 = \frac{3 x}{e} + 1 \]
    Point-slope form.✓ Proved
Answer \( y = \frac{3 x}{e} + 1 \)

✓ Nihil obstat Lines: 3 proved, 3 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 8.5

hard
Find the equation of the tangent line to \( \displaystyle x = \cos{\left(t \right)},\ y = t^{3} \) at \( \displaystyle t = \frac{\pi}{3} \).
  1. dy/dx = (dy/dt)/(dx/dt).
    Reviewed
  2. \[ \frac{d}{d t} \cos{\left(t \right)} = - \sin{\left(t \right)} \]
    dx/dt.✓ Proved
  3. \[ \frac{d}{d t} t^{3} = 3 t^{2} \]
    dy/dt.✓ Proved
  4. \[ - \frac{2 \sqrt{3} \pi^{2}}{9} \]
    The slope at t = \frac{\pi}{3}.Reviewed
  5. \[ \left[\begin{matrix}1 \cdot \frac{1}{2}\\\frac{\pi^{3}}{27}\end{matrix}\right] = \left[\begin{matrix}\frac{1}{2}\\\frac{\pi^{3}}{27}\end{matrix}\right] \]
    The point.Reviewed
  6. \[ - \frac{2 \sqrt{3} \pi^{2} \left(x - \frac{1}{2}\right)}{9} + \frac{\pi^{3}}{27} = - \frac{2 \sqrt{3} \pi^{2} x}{9} + \frac{\pi^{3}}{27} + \frac{\sqrt{3} \pi^{2}}{9} \]
    Point-slope form.✓ Proved
Answer \( y = - \frac{2 \sqrt{3} \pi^{2} x}{9} + \frac{\pi^{3}}{27} + \frac{\sqrt{3} \pi^{2}}{9} \)

✓ Nihil obstat Lines: 3 proved, 3 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 8.6

hard
Find the equation of the tangent line to \( \displaystyle x = e^{t},\ y = \sin{\left(t \right)} \) at \( \displaystyle t = 1 \).
  1. dy/dx = (dy/dt)/(dx/dt).
    Reviewed
  2. \[ \frac{d}{d t} e^{t} = e^{t} \]
    dx/dt.✓ Proved
  3. \[ \frac{d}{d t} \sin{\left(t \right)} = \cos{\left(t \right)} \]
    dy/dt.✓ Proved
  4. \[ \frac{\cos{\left(1 \right)}}{e} \]
    The slope at t = 1.Reviewed
  5. \[ \left[\begin{matrix}e\\\sin{\left(1 \right)}\end{matrix}\right] \]
    The point.Reviewed
  6. \[ \frac{\left(x - e\right) \cos{\left(1 \right)}}{e} + \sin{\left(1 \right)} = \frac{x \cos{\left(1 \right)}}{e} - \cos{\left(1 \right)} + \sin{\left(1 \right)} \]
    Point-slope form.✓ Proved
Answer \( y = \frac{x \cos{\left(1 \right)}}{e} - \cos{\left(1 \right)} + \sin{\left(1 \right)} \)

✓ Nihil obstat Lines: 3 proved, 3 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 8.7

hard
Find the equation of the tangent line to \( \displaystyle x = t^{3} - t,\ y = t^{3} \) at \( \displaystyle t = 1 \).
  1. dy/dx = (dy/dt)/(dx/dt).
    Reviewed
  2. \[ \frac{d}{d t} \left(t^{3} - t\right) = 3 t^{2} - 1 \]
    dx/dt.✓ Proved
  3. \[ \frac{d}{d t} t^{3} = 3 t^{2} \]
    dy/dt.✓ Proved
  4. \[ \frac{3}{2} \]
    The slope at t = 1.Reviewed
  5. \[ \left[\begin{matrix}0\\1\end{matrix}\right] \]
    The point.Reviewed
  6. \[ \frac{3 x}{2} + 1 \]
    Point-slope form.Reviewed
Answer \( y = \frac{3 x}{2} + 1 \)

✓ Nihil obstat Lines: 2 proved, 4 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 8.10

hard
Find the equation of the tangent line to \( \displaystyle x = 2 t,\ y = t^{3} \) at \( \displaystyle t = 2 \).
  1. dy/dx = (dy/dt)/(dx/dt).
    Reviewed
  2. \[ \frac{d}{d t} 2 t = 2 \]
    dx/dt.✓ Proved
  3. \[ \frac{d}{d t} t^{3} = 3 t^{2} \]
    dy/dt.✓ Proved
  4. \[ 6 \]
    The slope at t = 2.Reviewed
  5. \[ \left[\begin{matrix}4\\8\end{matrix}\right] \]
    The point.Reviewed
  6. \[ 6 x - 16 \]
    Point-slope form.Reviewed
Answer \( y = 6 x - 16 \)

✓ Nihil obstat Lines: 2 proved, 4 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 8.13

hard
Find the equation of the tangent line to \( \displaystyle x = t^{2} + 1,\ y = t^{3} \) at \( \displaystyle t = 2 \).
  1. dy/dx = (dy/dt)/(dx/dt).
    Reviewed
  2. \[ \frac{d}{d t} \left(t^{2} + 1\right) = 2 t \]
    dx/dt.✓ Proved
  3. \[ \frac{d}{d t} t^{3} = 3 t^{2} \]
    dy/dt.✓ Proved
  4. \[ 3 \]
    The slope at t = 2.Reviewed
  5. \[ \left[\begin{matrix}5\\8\end{matrix}\right] \]
    The point.Reviewed
  6. \[ 3 x - 7 \]
    Point-slope form.Reviewed
Answer \( y = 3 x - 7 \)

✓ Nihil obstat Lines: 2 proved, 4 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 8.14

hard
Find the equation of the tangent line to \( \displaystyle x = 2 t,\ y = 3 t + 1 \) at \( \displaystyle t = 1 \).
  1. dy/dx = (dy/dt)/(dx/dt).
    Reviewed
  2. \[ \frac{d}{d t} 2 t = 2 \]
    dx/dt.✓ Proved
  3. \[ \frac{d}{d t} \left(3 t + 1\right) = 3 \]
    dy/dt.✓ Proved
  4. \[ \frac{3}{2} \]
    The slope at t = 1.Reviewed
  5. \[ \left[\begin{matrix}2\\4\end{matrix}\right] \]
    The point.Reviewed
  6. \[ \frac{3 x}{2} + 1 \]
    Point-slope form.Reviewed
Answer \( y = \frac{3 x}{2} + 1 \)

✓ Nihil obstat Lines: 2 proved, 4 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 8.15

hard
Find the equation of the tangent line to \( \displaystyle x = \cos{\left(t \right)},\ y = \sin{\left(t \right)} \) at \( \displaystyle t = \frac{\pi}{3} \).
  1. dy/dx = (dy/dt)/(dx/dt).
    Reviewed
  2. \[ \frac{d}{d t} \cos{\left(t \right)} = - \sin{\left(t \right)} \]
    dx/dt.✓ Proved
  3. \[ \frac{d}{d t} \sin{\left(t \right)} = \cos{\left(t \right)} \]
    dy/dt.✓ Proved
  4. \[ \frac{\left(-1\right) \sqrt{3}}{3} = - \frac{\sqrt{3}}{3} \]
    The slope at t = \frac{\pi}{3}.Reviewed
  5. \[ \left[\begin{matrix}1 \cdot \frac{1}{2}\\\frac{\sqrt{3}}{2}\end{matrix}\right] = \left[\begin{matrix}\frac{1}{2}\\\frac{\sqrt{3}}{2}\end{matrix}\right] \]
    The point.Reviewed
  6. \[ - \frac{\sqrt{3} \left(x - \frac{1}{2}\right)}{3} + \frac{\sqrt{3}}{2} = - \frac{\sqrt{3} x}{3} + \frac{2 \sqrt{3}}{3} \]
    Point-slope form.✓ Proved
Answer \( y = - \frac{\sqrt{3} x}{3} + \frac{2 \sqrt{3}}{3} \)

✓ Nihil obstat Lines: 3 proved, 3 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 8.33

hard
Find the equation of the tangent line to \( \displaystyle x = t^{2} + 1,\ y = t^{2} - 3 t \) at \( \displaystyle t = 2 \).
  1. dy/dx = (dy/dt)/(dx/dt).
    Reviewed
  2. \[ \frac{d}{d t} \left(t^{2} + 1\right) = 2 t \]
    dx/dt.✓ Proved
  3. \[ \frac{d}{d t} \left(t^{2} - 3 t\right) = 2 t - 3 \]
    dy/dt.✓ Proved
  4. \[ 1 \cdot \frac{1}{4} = \frac{1}{4} \]
    The slope at t = 2.Reviewed
  5. \[ \left[\begin{matrix}5\\-2\end{matrix}\right] \]
    The point.Reviewed
  6. \[ \frac{x}{4} - \frac{13}{4} \]
    Point-slope form.Reviewed
Answer \( y = \frac{x}{4} - \frac{13}{4} \)

✓ Nihil obstat Lines: 2 proved, 4 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 8.34

hard
Find the equation of the tangent line to \( \displaystyle x = t^{2},\ y = t^{3} - 3 t \) at \( \displaystyle t = 2 \).
  1. dy/dx = (dy/dt)/(dx/dt).
    Reviewed
  2. \[ \frac{d}{d t} t^{2} = 2 t \]
    dx/dt.✓ Proved
  3. \[ \frac{d}{d t} \left(t^{3} - 3 t\right) = 3 t^{2} - 3 \]
    dy/dt.✓ Proved
  4. \[ \frac{9}{4} \]
    The slope at t = 2.Reviewed
  5. \[ \left[\begin{matrix}4\\2\end{matrix}\right] \]
    The point.Reviewed
  6. \[ \frac{9 x}{4} - 7 \]
    Point-slope form.Reviewed
Answer \( y = \frac{9 x}{4} - 7 \)

✓ Nihil obstat Lines: 2 proved, 4 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 8.35

hard
Find the equation of the tangent line to \( \displaystyle x = t^{2} + 1,\ y = 3 t + 1 \) at \( \displaystyle t = 1 \).
  1. dy/dx = (dy/dt)/(dx/dt).
    Reviewed
  2. \[ \frac{d}{d t} \left(t^{2} + 1\right) = 2 t \]
    dx/dt.✓ Proved
  3. \[ \frac{d}{d t} \left(3 t + 1\right) = 3 \]
    dy/dt.✓ Proved
  4. \[ \frac{3}{2} \]
    The slope at t = 1.Reviewed
  5. \[ \left[\begin{matrix}2\\4\end{matrix}\right] \]
    The point.Reviewed
  6. \[ \frac{3 x}{2} + 1 \]
    Point-slope form.Reviewed
Answer \( y = \frac{3 x}{2} + 1 \)

✓ Nihil obstat Lines: 2 proved, 4 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 8.37

hard
Find the equation of the tangent line to \( \displaystyle x = \cos{\left(t \right)},\ y = 3 t + 1 \) at \( \displaystyle t = \frac{\pi}{4} \).
  1. dy/dx = (dy/dt)/(dx/dt).
    Reviewed
  2. \[ \frac{d}{d t} \cos{\left(t \right)} = - \sin{\left(t \right)} \]
    dx/dt.✓ Proved
  3. \[ \frac{d}{d t} \left(3 t + 1\right) = 3 \]
    dy/dt.✓ Proved
  4. \[ - 3 \sqrt{2} \]
    The slope at t = \frac{\pi}{4}.Reviewed
  5. \[ \left[\begin{matrix}\frac{\sqrt{2}}{2}\\1 + \frac{3 \pi}{4}\end{matrix}\right] \]
    The point.Reviewed
  6. \[ - 3 \sqrt{2} \left(x - \frac{\sqrt{2}}{2}\right) + 1 + \frac{3 \pi}{4} = - 3 \sqrt{2} x + \frac{3 \pi}{4} + 4 \]
    Point-slope form.✓ Proved
Answer \( y = - 3 \sqrt{2} x + \frac{3 \pi}{4} + 4 \)

✓ Nihil obstat Lines: 3 proved, 3 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 8.40

hard
Find the equation of the tangent line to \( \displaystyle x = \cos{\left(t \right)},\ y = t^{2} - 3 t \) at \( \displaystyle t = \frac{\pi}{3} \).
  1. dy/dx = (dy/dt)/(dx/dt).
    Reviewed
  2. \[ \frac{d}{d t} \cos{\left(t \right)} = - \sin{\left(t \right)} \]
    dx/dt.✓ Proved
  3. \[ \frac{d}{d t} \left(t^{2} - 3 t\right) = 2 t - 3 \]
    dy/dt.✓ Proved
  4. \[ - \frac{2 \sqrt{3} \left(-3 + \frac{2 \pi}{3}\right)}{3} = \sqrt{3} \left(2 - \frac{4 \pi}{9}\right) \]
    The slope at t = \frac{\pi}{3}.✓ Proved
  5. \[ \left[\begin{matrix}1 \cdot \frac{1}{2}\\- \pi + \frac{\pi^{2}}{9}\end{matrix}\right] = \left[\begin{matrix}\frac{1}{2}\\- \pi + \frac{\pi^{2}}{9}\end{matrix}\right] \]
    The point.Reviewed
  6. \[ \sqrt{3} \left(2 - \frac{4 \pi}{9}\right) \left(x - \frac{1}{2}\right) - \pi + \frac{\pi^{2}}{9} = - \frac{4 \sqrt{3} \pi x}{9} + 2 \sqrt{3} x - \pi - \sqrt{3} + \frac{\pi^{2}}{9} + \frac{2 \sqrt{3} \pi}{9} \]
    Point-slope form.✓ Proved
Answer \( y = - \frac{4 \sqrt{3} \pi x}{9} + 2 \sqrt{3} x - \pi - \sqrt{3} + \frac{\pi^{2}}{9} + \frac{2 \sqrt{3} \pi}{9} \)

✓ Nihil obstat Lines: 4 proved, 2 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 8.42

hard
Find the equation of the tangent line to \( \displaystyle x = t^{2} + 1,\ y = t^{2} - 3 t \) at \( \displaystyle t = -1 \).
  1. dy/dx = (dy/dt)/(dx/dt).
    Reviewed
  2. \[ \frac{d}{d t} \left(t^{2} + 1\right) = 2 t \]
    dx/dt.✓ Proved
  3. \[ \frac{d}{d t} \left(t^{2} - 3 t\right) = 2 t - 3 \]
    dy/dt.✓ Proved
  4. \[ \frac{5}{2} \]
    The slope at t = -1.Reviewed
  5. \[ \left[\begin{matrix}2\\4\end{matrix}\right] \]
    The point.Reviewed
  6. \[ \frac{5 x}{2} - 1 \]
    Point-slope form.Reviewed
Answer \( y = \frac{5 x}{2} - 1 \)

✓ Nihil obstat Lines: 2 proved, 4 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 8.55

hard
Find the equation of the tangent line to \( \displaystyle x = t^{3} - t,\ y = 3 t + 1 \) at \( \displaystyle t = -1 \).
  1. dy/dx = (dy/dt)/(dx/dt).
    Reviewed
  2. \[ \frac{d}{d t} \left(t^{3} - t\right) = 3 t^{2} - 1 \]
    dx/dt.✓ Proved
  3. \[ \frac{d}{d t} \left(3 t + 1\right) = 3 \]
    dy/dt.✓ Proved
  4. \[ \frac{3}{2} \]
    The slope at t = -1.Reviewed
  5. \[ \left[\begin{matrix}0\\-2\end{matrix}\right] \]
    The point.Reviewed
  6. \[ \frac{3 x}{2} - 2 \]
    Point-slope form.Reviewed
Answer \( y = \frac{3 x}{2} - 2 \)

✓ Nihil obstat Lines: 2 proved, 4 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 8.60

hard
Find the equation of the tangent line to \( \displaystyle x = e^{t},\ y = t^{2} - 3 t \) at \( \displaystyle t = 1 \).
  1. dy/dx = (dy/dt)/(dx/dt).
    Reviewed
  2. \[ \frac{d}{d t} e^{t} = e^{t} \]
    dx/dt.✓ Proved
  3. \[ \frac{d}{d t} \left(t^{2} - 3 t\right) = 2 t - 3 \]
    dy/dt.✓ Proved
  4. \[ - \frac{1}{e} \]
    The slope at t = 1.Reviewed
  5. \[ \left[\begin{matrix}e\\-2\end{matrix}\right] \]
    The point.Reviewed
  6. \[ \frac{e - x}{e} - 2 = - \frac{x}{e} - 1 \]
    Point-slope form.✓ Proved
Answer \( y = - \frac{x}{e} - 1 \)

✓ Nihil obstat Lines: 3 proved, 3 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 8.64

hard
Find the equation of the tangent line to \( \displaystyle x = 2 t,\ y = t^{3} \) at \( \displaystyle t = -1 \).
  1. dy/dx = (dy/dt)/(dx/dt).
    Reviewed
  2. \[ \frac{d}{d t} 2 t = 2 \]
    dx/dt.✓ Proved
  3. \[ \frac{d}{d t} t^{3} = 3 t^{2} \]
    dy/dt.✓ Proved
  4. \[ \frac{3}{2} \]
    The slope at t = -1.Reviewed
  5. \[ \left[\begin{matrix}-2\\-1\end{matrix}\right] \]
    The point.Reviewed
  6. \[ \frac{3 x}{2} + 2 \]
    Point-slope form.Reviewed
Answer \( y = \frac{3 x}{2} + 2 \)

✓ Nihil obstat Lines: 2 proved, 4 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 tangent lines to parametric curves practice

Find the equation of the tangent line to \( \displaystyle x = e^{t},\ y = t^{3} - 3 t \) at \( \displaystyle t = -1 \).
Problem 8.9medium✓ Every equation proved
Find the equation of the tangent line to \( \displaystyle x = t^{2} + 1,\ y = t^{3} - 3 t \) at \( \displaystyle t = -1 \).
Problem 8.12medium✓ Every equation proved
Find the equation of the tangent line to \( \displaystyle x = t^{2},\ y = t^{3} - 3 t \) at \( \displaystyle t = -1 \).
Problem 8.43medium✓ Every equation proved
Find the equation of the tangent line to \( \displaystyle x = 2 t,\ y = t^{3} - 3 t \) at \( \displaystyle t = -1 \).
Problem 8.67medium✓ Every equation proved
Find the equation of the tangent line to \( \displaystyle x = t^{3} - t,\ y = t^{3} - 3 t \) at \( \displaystyle t = -1 \).
Problem 8.90medium✓ Every equation proved
Find the equation of the tangent line to \( \displaystyle x = e^{t},\ y = \sin{\left(t \right)} \) at \( \displaystyle t = -1 \).
Problem 8.8hard✓ Every equation proved
Find the equation of the tangent line to \( \displaystyle x = 2 t,\ y = \sin{\left(t \right)} \) at \( \displaystyle t = -1 \).
Problem 8.11hard✓ Every equation proved
Find the equation of the tangent line to \( \displaystyle x = 2 t,\ y = t^{3} \) at \( \displaystyle t = 1 \).
Problem 8.31hard✓ Nihil obstat
Find the equation of the tangent line to \( \displaystyle x = t^{2},\ y = t^{3} - 3 t \) at \( \displaystyle t = 1 \).
Problem 8.32hard✓ Every equation proved
Find the equation of the tangent line to \( \displaystyle x = e^{t},\ y = t^{3} - 3 t \) at \( \displaystyle t = 1 \).
Problem 8.36hard✓ Every equation proved
Find the equation of the tangent line to \( \displaystyle x = \cos{\left(t \right)},\ y = 3 t + 1 \) at \( \displaystyle t = \frac{\pi}{3} \).
Problem 8.38hard✓ Nihil obstat
Find the equation of the tangent line to \( \displaystyle x = e^{t},\ y = t^{3} \) at \( \displaystyle t = -1 \).
Problem 8.39hard✓ Every equation proved
Find the equation of the tangent line to \( \displaystyle x = t^{3} - t,\ y = t^{2} - 3 t \) at \( \displaystyle t = 2 \).
Problem 8.41hard✓ Nihil obstat
Find the equation of the tangent line to \( \displaystyle x = t^{2},\ y = t^{2} - 3 t \) at \( \displaystyle t = 1 \).
Problem 8.44hard✓ Nihil obstat
Find the equation of the tangent line to \( \displaystyle x = \cos{\left(t \right)},\ y = t^{3} \) at \( \displaystyle t = \frac{\pi}{4} \).
Problem 8.45hard✓ Nihil obstat
Find the equation of the tangent line to \( \displaystyle x = 2 t,\ y = t^{3} - 3 t \) at \( \displaystyle t = 1 \).
Problem 8.51hard✓ Every equation proved
Find the equation of the tangent line to \( \displaystyle x = \cos{\left(t \right)},\ y = t^{2} - 3 t \) at \( \displaystyle t = \frac{\pi}{6} \).
Problem 8.52hard✓ Nihil obstat
Find the equation of the tangent line to \( \displaystyle x = 2 t,\ y = t^{3} - 3 t \) at \( \displaystyle t = 2 \).
Problem 8.53hard✓ Nihil obstat
Find the equation of the tangent line to \( \displaystyle x = t^{2} + 1,\ y = t^{3} - 3 t \) at \( \displaystyle t = 1 \).
Problem 8.54hard✓ Every equation proved
Find the equation of the tangent line to \( \displaystyle x = t^{3} - t,\ y = \sin{\left(t \right)} \) at \( \displaystyle t = 1 \).
Problem 8.56hard✓ Every equation proved
Find the equation of the tangent line to \( \displaystyle x = \cos{\left(t \right)},\ y = t^{3} - 3 t \) at \( \displaystyle t = \frac{\pi}{6} \).
Problem 8.57hard✓ Nihil obstat
Find the equation of the tangent line to \( \displaystyle x = t^{2} + 1,\ y = t^{2} - 3 t \) at \( \displaystyle t = 1 \).
Problem 8.58hard✓ Nihil obstat
Find the equation of the tangent line to \( \displaystyle x = t^{3} - t,\ y = \sin{\left(t \right)} \) at \( \displaystyle t = -1 \).
Problem 8.59hard✓ Every equation proved
Find the equation of the tangent line to \( \displaystyle x = e^{t},\ y = 3 t + 1 \) at \( \displaystyle t = 2 \).
Problem 8.61hard✓ Nihil obstat
Find the equation of the tangent line to \( \displaystyle x = t^{2},\ y = 3 t + 1 \) at \( \displaystyle t = -1 \).
Problem 8.62hard✓ Every equation proved
Find the equation of the tangent line to \( \displaystyle x = t^{2},\ y = t^{2} - 3 t \) at \( \displaystyle t = 2 \).
Problem 8.63hard✓ Nihil obstat
Find the equation of the tangent line to \( \displaystyle x = 2 t,\ y = \sin{\left(t \right)} \) at \( \displaystyle t = 2 \).
Problem 8.65hard✓ Nihil obstat
Find the equation of the tangent line to \( \displaystyle x = \cos{\left(t \right)},\ y = t^{3} - 3 t \) at \( \displaystyle t = \frac{\pi}{3} \).
Problem 8.66hard✓ Every equation proved
Find the equation of the tangent line to \( \displaystyle x = t^{2},\ y = \sin{\left(t \right)} \) at \( \displaystyle t = 1 \).
Problem 8.68hard✓ Every equation proved
Find the equation of the tangent line to \( \displaystyle x = t^{2} + 1,\ y = 3 t + 1 \) at \( \displaystyle t = -1 \).
Problem 8.69hard✓ Every equation proved
Find the equation of the tangent line to \( \displaystyle x = t^{2},\ y = \sin{\left(t \right)} \) at \( \displaystyle t = -1 \).
Problem 8.70hard✓ Nihil obstat
Find the equation of the tangent line to \( \displaystyle x = t^{2} + 1,\ y = \sin{\left(t \right)} \) at \( \displaystyle t = 2 \).
Problem 8.71hard✓ Nihil obstat
Find the equation of the tangent line to \( \displaystyle x = t^{3} - t,\ y = \sin{\left(t \right)} \) at \( \displaystyle t = 2 \).
Problem 8.72hard✓ Every equation proved
Find the equation of the tangent line to \( \displaystyle x = e^{t},\ y = 3 t + 1 \) at \( \displaystyle t = -1 \).
Problem 8.73hard✓ Every equation proved
Find the equation of the tangent line to \( \displaystyle x = e^{t},\ y = t^{3} \) at \( \displaystyle t = 2 \).
Problem 8.74hard✓ Nihil obstat
Find the equation of the tangent line to \( \displaystyle x = t^{3} - t,\ y = t^{2} - 3 t \) at \( \displaystyle t = -1 \).
Problem 8.75hard✓ Nihil obstat
Find the equation of the tangent line to \( \displaystyle x = e^{t},\ y = t^{2} - 3 t \) at \( \displaystyle t = 2 \).
Problem 8.76hard✓ Every equation proved
Find the equation of the tangent line to \( \displaystyle x = 2 t,\ y = t^{2} - 3 t \) at \( \displaystyle t = 2 \).
Problem 8.77hard✓ Every equation proved
Find the equation of the tangent line to \( \displaystyle x = t^{3} - t,\ y = t^{3} \) at \( \displaystyle t = -1 \).
Problem 8.78hard✓ Nihil obstat
Find the equation of the tangent line to \( \displaystyle x = t^{2} + 1,\ y = t^{3} - 3 t \) at \( \displaystyle t = 2 \).
Problem 8.79hard✓ Nihil obstat
Find the equation of the tangent line to \( \displaystyle x = t^{2} + 1,\ y = \sin{\left(t \right)} \) at \( \displaystyle t = -1 \).
Problem 8.80hard✓ Every equation proved
Find the equation of the tangent line to \( \displaystyle x = 2 t,\ y = 3 t + 1 \) at \( \displaystyle t = 2 \).
Problem 8.81hard✓ Every equation proved
Find the equation of the tangent line to \( \displaystyle x = t^{2},\ y = t^{3} \) at \( \displaystyle t = -1 \).
Problem 8.82hard✓ Every equation proved
Find the equation of the tangent line to \( \displaystyle x = t^{2} + 1,\ y = t^{3} \) at \( \displaystyle t = -1 \).
Problem 8.83hard✓ Every equation proved
Find the equation of the tangent line to \( \displaystyle x = \cos{\left(t \right)},\ y = \sin{\left(t \right)} \) at \( \displaystyle t = \frac{\pi}{6} \).
Problem 8.84hard✓ Nihil obstat
Find the equation of the tangent line to \( \displaystyle x = t^{2},\ y = 3 t + 1 \) at \( \displaystyle t = 1 \).
Problem 8.85hard✓ Nihil obstat
Find the equation of the tangent line to \( \displaystyle x = t^{2},\ y = t^{3} \) at \( \displaystyle t = 1 \).
Problem 8.86hard✓ Every equation proved
Find the equation of the tangent line to \( \displaystyle x = t^{3} - t,\ y = 3 t + 1 \) at \( \displaystyle t = 1 \).
Problem 8.87hard✓ Nihil obstat
Find the equation of the tangent line to \( \displaystyle x = t^{3} - t,\ y = t^{3} \) at \( \displaystyle t = 2 \).
Problem 8.88hard✓ Nihil obstat
Find the equation of the tangent line to \( \displaystyle x = t^{2} + 1,\ y = t^{3} \) at \( \displaystyle t = 1 \).
Problem 8.89hard✓ Every equation proved