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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. 30 problems with worked solutions; in 30 of them every equation is proved by a computer algebra system.

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 \( \displaystyle x = e^{t},\ y = t^{3} \) at \( \displaystyle t = 1 \).

Problem 8.1hard✓ Every equation proved

Find the equation of the tangent line to \( \displaystyle x = t^{3} - t,\ y = t^{3} - 3 t \) at \( \displaystyle t = 2 \).

Problem 8.2hard✓ Every equation proved

Find the equation of the tangent line to \( \displaystyle x = 2 t,\ y = t^{2} - 3 t \) at \( \displaystyle t = 1 \).

Problem 8.3hard✓ Every equation proved

Find the equation of the tangent line to \( \displaystyle x = e^{t},\ y = 3 t + 1 \) at \( \displaystyle t = 1 \).

Problem 8.4hard✓ Every equation proved

Find the equation of the tangent line to \( \displaystyle x = \cos{\left(t \right)},\ y = t^{3} \) at \( \displaystyle t = \frac{\pi}{3} \).

Problem 8.5hard✓ 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 8.6hard✓ Every equation proved

Find the equation of the tangent line to \( \displaystyle x = t^{3} - t,\ y = t^{3} \) at \( \displaystyle t = 1 \).

Problem 8.7hard✓ 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 \( \displaystyle x = 2 t,\ y = t^{3} \) at \( \displaystyle t = 2 \).

Problem 8.10hard✓ 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 \( \displaystyle x = t^{2} + 1,\ y = t^{3} \) at \( \displaystyle t = 2 \).

Problem 8.13hard✓ Every equation proved

Find the equation of the tangent line to \( \displaystyle x = 2 t,\ y = 3 t + 1 \) at \( \displaystyle t = 1 \).

Problem 8.14hard✓ Every equation proved

Find the equation of the tangent line to \( \displaystyle x = \cos{\left(t \right)},\ y = \sin{\left(t \right)} \) at \( \displaystyle t = \frac{\pi}{3} \).

Problem 8.15hard✓ 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✓ 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 8.32hard✓ Every equation proved

Find the equation of the tangent line to \( \displaystyle x = t^{2} + 1,\ y = t^{2} - 3 t \) at \( \displaystyle t = 2 \).

Problem 8.33hard✓ Every equation proved

Find the equation of the tangent line to \( \displaystyle x = t^{2},\ y = t^{3} - 3 t \) at \( \displaystyle t = 2 \).

Problem 8.34hard✓ 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 8.35hard✓ 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 \( \displaystyle t = \frac{\pi}{4} \).

Problem 8.37hard✓ 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✓ Every equation proved

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 \( \displaystyle x = \cos{\left(t \right)},\ y = t^{2} - 3 t \) at \( \displaystyle t = \frac{\pi}{3} \).

Problem 8.40hard✓ 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✓ Every equation proved

Find the equation of the tangent line to \( \displaystyle x = t^{2} + 1,\ y = t^{2} - 3 t \) at \( \displaystyle t = -1 \).

Problem 8.42hard✓ Every equation proved

Find the equation of the tangent line to \( \displaystyle x = t^{2},\ y = t^{2} - 3 t \) at \( \displaystyle t = 1 \).

Problem 8.44hard✓ Every equation proved

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✓ Every equation proved