Second Derivatives of Parametric Equations - AP Calculus BC

1. What is $\frac{dy}{dx}$ for $y(t) = t^2$ and $x(t) = t^2$?

   $1$. Since both functions are identical, $\frac{dy}{dx} = \frac{2t}{2t} = 1$.

2. Find $\frac{dx}{dt}$ for $x(t) = 2t^2 + 3t$.

   $4t + 3$. Apply power rule: derivative of $2t^2$ is $4t$, derivative of $3t$ is $3$.

3. Calculate $\frac{dy}{dx}$ when $y(t) = t^2 + 1$ and $x(t) = 3t$.

   $\frac{2t}{3}$. Calculate: $\frac{dy}{dt} = 2t$ and $\frac{dx}{dt} = 3$, so $\frac{dy}{dx} = \frac{2t}{3}$.

4. State the formula for the second derivative in parametric form.

   $\frac{d^2y}{dx^2} = \frac{\frac{d}{dt}(\frac{dy}{dx})}{\frac{dx}{dt}}$. Apply the chain rule: differentiate $\frac{dy}{dx}$ with respect to $t$, then divide by $\frac{dx}{dt}$.

5. What is $\frac{dy}{dt}$ if $y(t) = \ln(t)$?

   $\frac{1}{t}$. The derivative of $\ln(t)$ with respect to $t$ is $\frac{1}{t}$.

6. What is $\frac{dy}{dx}$ for $y(t) = e^t$ and $x(t) = t^2$?

   $\frac{e^t}{2t}$. Calculate: $\frac{dy}{dt} = e^t$ and $\frac{dx}{dt} = 2t$, so $\frac{dy}{dx} = \frac{e^t}{2t}$.

7. Find $\frac{dx}{dt}$ if $x(t) = t^3 - 4t^2$.

   $3t^2 - 8t$. Apply power rule: derivative of $t^3$ is $3t^2$, derivative of $-4t^2$ is $-8t$.

8. What is the expression for $\frac{dy}{dx}$ if $y(t) = t^3$ and $x(t) = t$?

   $3t^2$. Calculate: $\frac{dy}{dt} = 3t^2$ and $\frac{dx}{dt} = 1$, so $\frac{dy}{dx} = 3t^2$.

9. What is $\frac{dy}{dx}$ for $y(t) = \tan(t)$ and $x(t) = t$?

   $\sec^2(t)$. Calculate: $\frac{dy}{dt} = \sec^2(t)$ and $\frac{dx}{dt} = 1$, so $\frac{dy}{dx} = \sec^2(t)$.

10. Identify the formula for $\frac{dy}{dx}$ in terms of $x(t)$ and $y(t)$.

    $\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}$. Use the chain rule to express the slope in terms of the parameter $t$.

11. Find the expression for $\frac{dy}{dx}$ given $y(t) = \sin(t)$ and $x(t) = \cos(t)$.

    $-\frac{\cos(t)}{\sin(t)}$. Calculate: $\frac{dy}{dt} = \cos(t)$ and $\frac{dx}{dt} = -\sin(t)$, so $\frac{dy}{dx} = -\cot(t)$.

12. Calculate $\frac{dy}{dx}$ for $y(t) = 5t - t^2$ and $x(t) = t^3$.

    $\frac{5 - 2t}{3t^2}$. Calculate: $\frac{dy}{dt} = 5-2t$ and $\frac{dx}{dt} = 3t^2$, so $\frac{dy}{dx} = \frac{5-2t}{3t^2}$.

13. What is the derivative $\frac{dy}{dt}$ for $y(t) = t^2 - 5t + 6$?

    $2t - 5$. Apply power rule: derivative of $t^2$ is $2t$, derivative of $-5t$ is $-5$.

14. What is $\frac{dx}{dt}$ for $x(t) = \tan(t)$?

    $\sec^2(t)$. The derivative of $\tan(t)$ with respect to $t$ is $\sec^2(t)$.

15. Identify $\frac{dx}{dt}$ if $x(t) = e^t$.

    $e^t$. The derivative of $e^t$ with respect to $t$ is $e^t$.

16. State the expression for $\frac{dy}{dx}$ if $y(t) = \cos(t)$ and $x(t) = \sin(t)$.

    $-\cot(t)$. Same calculation as earlier: $\frac{dy}{dx} = -\cot(t)$.

17. What is the derivative $\frac{dy}{dt}$ for $y(t) = t^3 - 4t$?

    $3t^2 - 4$. Apply power rule: derivative of $t^3$ is $3t^2$, derivative of $-4t$ is $-4$.

18. Identify $\frac{dy}{dx}$ if $y(t) = e^t$ and $x(t) = e^{-t}$.

    $ -$e^{2t}$ $. Calculate: $\frac{dy}{dt} = e^t$ and $\frac{dx}{dt} = -e^{-t}$, so $\frac{dy}{dx} = -e^{2t}$

19. Identify $\frac{dy}{dx}$ for $y(t) = t^2 + 3t$ and $x(t) = 2t$.

    $\frac{2t + 3}{2}$. Calculate: $\frac{dy}{dt} = 2t+3$ and $\frac{dx}{dt} = 2$, so $\frac{dy}{dx} = \frac{2t+3}{2}$.

20. State the second derivative formula in terms of $t$ for $x(t)$ and $y(t)$.

    $\frac{d^2y}{dx^2} = \frac{d}{dt}(\frac{dy}{dx}) \cdot \frac{1}{\frac{dx}{dt}}$. Alternative form of the second derivative formula using the reciprocal of $\frac{dx}{dt}$.

Study these flashcards