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AP Calculus BC Flashcards: Second Derivatives Of Parametric Equations

Study Second Derivatives Of Parametric Equations in AP Calculus BC with focused flashcards that help you recognize the idea, recall the key rule, and apply it in practice-style prompts.

← Back to flashcard decks

What this deck covers

This deck focuses on Second Derivatives Of Parametric Equations, giving you a quick way to review the definitions, rules, and examples that matter most for AP Calculus BC.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

AP Calculus BC Flashcards: Second Derivatives Of Parametric Equations

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QUESTION

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

Tap or drag to reveal answer

ANSWER

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

Swipe Right = I Know It! 🎉

Swipe Left = Still Learning

All flashcards

Flashcard 1: What is the derivative $\frac{dy}{dt}$ for $y(t) = t^3 - 4t$ ?

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

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

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

Flashcard 3: What is $\frac{dy}{dx}$ for $y(t) = e^t$ and $x(t) = t^2$ ?

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

Flashcard 4: What is $\frac{dx}{dt}$ for $x(t) = \tan(t)$ ?

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

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

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

Flashcard 6: Identify $\frac{dy}{dx}$ for $y(t) = t^2 + 3t$ and $x(t) = 2t$ .

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

Flashcard 7: Identify the formula for $\frac{dy}{dx}$ in terms of $x(t)$ and $y(t)$ .

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

Flashcard 8: Identify $\frac{dx}{dt}$ if $x(t) = e^t$ .

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

Flashcard 9: What is $\frac{dy}{dx}$ for $y(t) = t^2$ and $x(t) = t^2$ ?

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

Flashcard 10: What is the derivative $\frac{dy}{dt}$ for $y(t) = t^2 - 5t + 6$ ?

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

Flashcard 11: What is the expression for $\frac{dy}{dx}$ if $y(t) = t^3$ and $x(t) = t$ ?

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

Flashcard 12: Find $\frac{dx}{dt}$ if $x(t) = t^3 - 4t^2$ .

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

Flashcard 13: State the second derivative formula in terms of $t$ for $x(t)$ and $y(t)$ .

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

Flashcard 14: Find $\frac{dx}{dt}$ for $x(t) = 2t^2 + 3t$ .

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

Flashcard 15: Find the expression for $\frac{dy}{dx}$ given $y(t) = \sin(t)$ and $x(t) = \cos(t)$ .

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

Flashcard 16: Identify $\frac{dy}{dx}$ if $y(t) = e^t$ and $x(t) = e^{-t}$ .

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

Flashcard 17: What is $\frac{dy}{dx}$ for $y(t) = \tan(t)$ and $x(t) = t$ ?

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

Flashcard 18: State the expression for $\frac{dy}{dx}$ if $y(t) = \cos(t)$ and $x(t) = \sin(t)$ .

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

Flashcard 19: Calculate $\frac{dy}{dx}$ when $y(t) = t^2 + 1$ and $x(t) = 3t$ .

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

Flashcard 20: State the formula for the second derivative in parametric form.

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

Flashcard 21: Find $\frac{dx}{dt}$ if $x(t) = t^3 - 4t^2$ .

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

Flashcard 22: Identify $\frac{dy}{dx}$ for $y(t) = t^2 + 3t$ and $x(t) = 2t$ .

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

Flashcard 23: What is $\frac{dy}{dx}$ for $y(t) = t^2$ and $x(t) = t^2$ ?

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

Flashcard 24: What is the derivative $\frac{dy}{dt}$ for $y(t) = t^2 - 5t + 6$ ?

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

Flashcard 25: What is the expression for $\frac{dy}{dx}$ if $y(t) = t^3$ and $x(t) = t$ ?

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

Flashcard 26: What is $\frac{dx}{dt}$ for $x(t) = \tan(t)$ ?

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

Flashcard 27: What is $\frac{dy}{dx}$ for $y(t) = e^t$ and $x(t) = t^2$ ?

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

Flashcard 28: State the second derivative formula in terms of $t$ for $x(t)$ and $y(t)$ .

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

Flashcard 29: Find $\frac{dx}{dt}$ for $x(t) = 2t^2 + 3t$ .

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

Flashcard 30: Find the expression for $\frac{dy}{dx}$ given $y(t) = \sin(t)$ and $x(t) = \cos(t)$ .

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

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AP Calculus BC Flashcards: Second Derivatives Of Parametric Equations

Study Second Derivatives Of Parametric Equations in AP Calculus BC with focused flashcards that help you recognize the idea, recall the key rule, and apply it in practice-style prompts.

← Back to flashcard decks

What this deck covers

This deck focuses on Second Derivatives Of Parametric Equations, giving you a quick way to review the definitions, rules, and examples that matter most for AP Calculus BC.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

AP Calculus BC Flashcards: Second Derivatives Of Parametric Equations

/ 30

0 reviewed

0% Complete

0 reviewing

QUESTION

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

Tap or drag to reveal answer

ANSWER

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

Swipe Right = I Know It! 🎉

Swipe Left = Still Learning

All flashcards

Flashcard 1: What is the derivative $\frac{dy}{dt}$ for $y(t) = t^3 - 4t$ ?

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

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

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

Flashcard 3: What is $\frac{dy}{dx}$ for $y(t) = e^t$ and $x(t) = t^2$ ?

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

Flashcard 4: What is $\frac{dx}{dt}$ for $x(t) = \tan(t)$ ?

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

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

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

Flashcard 6: Identify $\frac{dy}{dx}$ for $y(t) = t^2 + 3t$ and $x(t) = 2t$ .

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

Flashcard 7: Identify the formula for $\frac{dy}{dx}$ in terms of $x(t)$ and $y(t)$ .

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

Flashcard 8: Identify $\frac{dx}{dt}$ if $x(t) = e^t$ .

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

Flashcard 9: What is $\frac{dy}{dx}$ for $y(t) = t^2$ and $x(t) = t^2$ ?

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

Flashcard 10: What is the derivative $\frac{dy}{dt}$ for $y(t) = t^2 - 5t + 6$ ?

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

Flashcard 11: What is the expression for $\frac{dy}{dx}$ if $y(t) = t^3$ and $x(t) = t$ ?

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

Flashcard 12: Find $\frac{dx}{dt}$ if $x(t) = t^3 - 4t^2$ .

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

Flashcard 13: State the second derivative formula in terms of $t$ for $x(t)$ and $y(t)$ .

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

Flashcard 14: Find $\frac{dx}{dt}$ for $x(t) = 2t^2 + 3t$ .

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

Flashcard 15: Find the expression for $\frac{dy}{dx}$ given $y(t) = \sin(t)$ and $x(t) = \cos(t)$ .

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

Flashcard 16: Identify $\frac{dy}{dx}$ if $y(t) = e^t$ and $x(t) = e^{-t}$ .

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

Flashcard 17: What is $\frac{dy}{dx}$ for $y(t) = \tan(t)$ and $x(t) = t$ ?

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

Flashcard 18: State the expression for $\frac{dy}{dx}$ if $y(t) = \cos(t)$ and $x(t) = \sin(t)$ .

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

Flashcard 19: Calculate $\frac{dy}{dx}$ when $y(t) = t^2 + 1$ and $x(t) = 3t$ .

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

Flashcard 20: State the formula for the second derivative in parametric form.

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

Flashcard 21: Find $\frac{dx}{dt}$ if $x(t) = t^3 - 4t^2$ .

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

Flashcard 22: Identify $\frac{dy}{dx}$ for $y(t) = t^2 + 3t$ and $x(t) = 2t$ .

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

Flashcard 23: What is $\frac{dy}{dx}$ for $y(t) = t^2$ and $x(t) = t^2$ ?

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

Flashcard 24: What is the derivative $\frac{dy}{dt}$ for $y(t) = t^2 - 5t + 6$ ?

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

Flashcard 25: What is the expression for $\frac{dy}{dx}$ if $y(t) = t^3$ and $x(t) = t$ ?

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

Flashcard 26: What is $\frac{dx}{dt}$ for $x(t) = \tan(t)$ ?

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

Flashcard 27: What is $\frac{dy}{dx}$ for $y(t) = e^t$ and $x(t) = t^2$ ?

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

Flashcard 28: State the second derivative formula in terms of $t$ for $x(t)$ and $y(t)$ .

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

Flashcard 29: Find $\frac{dx}{dt}$ for $x(t) = 2t^2 + 3t$ .

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

Flashcard 30: Find the expression for $\frac{dy}{dx}$ given $y(t) = \sin(t)$ and $x(t) = \cos(t)$ .

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

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AP Calculus BC Flashcards: Second Derivatives Of Parametric Equations

What this deck covers

How to use these flashcards

AP Calculus BC Flashcards: Second Derivatives Of Parametric Equations

All flashcards

Flashcard 1: What is the derivative dydt\frac{dy}{dt}dtdy​ for y(t)=t3−4ty(t) = t^3 - 4ty(t)=t3−4t?

Flashcard 2: Calculate dydx\frac{dy}{dx}dxdy​ for y(t)=5t−t2y(t) = 5t - t^2y(t)=5t−t2 and x(t)=t3x(t) = t^3x(t)=t3.

Flashcard 3: What is dydx\frac{dy}{dx}dxdy​ for y(t)=ety(t) = e^ty(t)=et and x(t)=t2x(t) = t^2x(t)=t2?

Flashcard 4: What is dxdt\frac{dx}{dt}dtdx​ for x(t)=tan⁡(t)x(t) = \tan(t)x(t)=tan(t)?

Flashcard 5: What is dydt\frac{dy}{dt}dtdy​ if y(t)=ln⁡(t)y(t) = \ln(t)y(t)=ln(t)?

Flashcard 6: Identify dydx\frac{dy}{dx}dxdy​ for y(t)=t2+3ty(t) = t^2 + 3ty(t)=t2+3t and x(t)=2tx(t) = 2tx(t)=2t.

Flashcard 7: Identify the formula for dydx\frac{dy}{dx}dxdy​ in terms of x(t)x(t)x(t) and y(t)y(t)y(t).

Flashcard 8: Identify dxdt\frac{dx}{dt}dtdx​ if x(t)=etx(t) = e^tx(t)=et.

Flashcard 9: What is dydx\frac{dy}{dx}dxdy​ for y(t)=t2y(t) = t^2y(t)=t2 and x(t)=t2x(t) = t^2x(t)=t2?

Flashcard 10: What is the derivative dydt\frac{dy}{dt}dtdy​ for y(t)=t2−5t+6y(t) = t^2 - 5t + 6y(t)=t2−5t+6?

Flashcard 11: What is the expression for dydx\frac{dy}{dx}dxdy​ if y(t)=t3y(t) = t^3y(t)=t3 and x(t)=tx(t) = tx(t)=t?

Flashcard 12: Find dxdt\frac{dx}{dt}dtdx​ if x(t)=t3−4t2x(t) = t^3 - 4t^2x(t)=t3−4t2.

Flashcard 13: State the second derivative formula in terms of ttt for x(t)x(t)x(t) and y(t)y(t)y(t).

Flashcard 14: Find dxdt\frac{dx}{dt}dtdx​ for x(t)=2t2+3tx(t) = 2t^2 + 3tx(t)=2t2+3t.

Flashcard 15: Find the expression for dydx\frac{dy}{dx}dxdy​ given y(t)=sin⁡(t)y(t) = \sin(t)y(t)=sin(t) and x(t)=cos⁡(t)x(t) = \cos(t)x(t)=cos(t).

Flashcard 16: Identify dydx\frac{dy}{dx}dxdy​ if y(t)=ety(t) = e^ty(t)=et and x(t)=e−tx(t) = e^{-t}x(t)=e−t.

Flashcard 17: What is dydx\frac{dy}{dx}dxdy​ for y(t)=tan⁡(t)y(t) = \tan(t)y(t)=tan(t) and x(t)=tx(t) = tx(t)=t?

Flashcard 18: State the expression for dydx\frac{dy}{dx}dxdy​ if y(t)=cos⁡(t)y(t) = \cos(t)y(t)=cos(t) and x(t)=sin⁡(t)x(t) = \sin(t)x(t)=sin(t).

Flashcard 19: Calculate dydx\frac{dy}{dx}dxdy​ when y(t)=t2+1y(t) = t^2 + 1y(t)=t2+1 and x(t)=3tx(t) = 3tx(t)=3t.

Flashcard 20: State the formula for the second derivative in parametric form.

Flashcard 21: Find dxdt\frac{dx}{dt}dtdx​ if x(t)=t3−4t2x(t) = t^3 - 4t^2x(t)=t3−4t2.

Flashcard 22: Identify dydx\frac{dy}{dx}dxdy​ for y(t)=t2+3ty(t) = t^2 + 3ty(t)=t2+3t and x(t)=2tx(t) = 2tx(t)=2t.

Flashcard 23: What is dydx\frac{dy}{dx}dxdy​ for y(t)=t2y(t) = t^2y(t)=t2 and x(t)=t2x(t) = t^2x(t)=t2?

Flashcard 24: What is the derivative dydt\frac{dy}{dt}dtdy​ for y(t)=t2−5t+6y(t) = t^2 - 5t + 6y(t)=t2−5t+6?

Flashcard 25: What is the expression for dydx\frac{dy}{dx}dxdy​ if y(t)=t3y(t) = t^3y(t)=t3 and x(t)=tx(t) = tx(t)=t?

Flashcard 26: What is dxdt\frac{dx}{dt}dtdx​ for x(t)=tan⁡(t)x(t) = \tan(t)x(t)=tan(t)?

Flashcard 27: What is dydx\frac{dy}{dx}dxdy​ for y(t)=ety(t) = e^ty(t)=et and x(t)=t2x(t) = t^2x(t)=t2?

Flashcard 28: State the second derivative formula in terms of ttt for x(t)x(t)x(t) and y(t)y(t)y(t).

Flashcard 29: Find dxdt\frac{dx}{dt}dtdx​ for x(t)=2t2+3tx(t) = 2t^2 + 3tx(t)=2t2+3t.

Flashcard 30: Find the expression for dydx\frac{dy}{dx}dxdy​ given y(t)=sin⁡(t)y(t) = \sin(t)y(t)=sin(t) and x(t)=cos⁡(t)x(t) = \cos(t)x(t)=cos(t).

Opening subject page...

AP Calculus BC Flashcards: Second Derivatives Of Parametric Equations

What this deck covers

How to use these flashcards

AP Calculus BC Flashcards: Second Derivatives Of Parametric Equations

All flashcards

Flashcard 1: What is the derivative dydt\frac{dy}{dt}dtdy​ for y(t)=t3−4ty(t) = t^3 - 4ty(t)=t3−4t?

Flashcard 2: Calculate dydx\frac{dy}{dx}dxdy​ for y(t)=5t−t2y(t) = 5t - t^2y(t)=5t−t2 and x(t)=t3x(t) = t^3x(t)=t3.

Flashcard 3: What is dydx\frac{dy}{dx}dxdy​ for y(t)=ety(t) = e^ty(t)=et and x(t)=t2x(t) = t^2x(t)=t2?

Flashcard 4: What is dxdt\frac{dx}{dt}dtdx​ for x(t)=tan⁡(t)x(t) = \tan(t)x(t)=tan(t)?

Flashcard 5: What is dydt\frac{dy}{dt}dtdy​ if y(t)=ln⁡(t)y(t) = \ln(t)y(t)=ln(t)?

Flashcard 6: Identify dydx\frac{dy}{dx}dxdy​ for y(t)=t2+3ty(t) = t^2 + 3ty(t)=t2+3t and x(t)=2tx(t) = 2tx(t)=2t.

Flashcard 7: Identify the formula for dydx\frac{dy}{dx}dxdy​ in terms of x(t)x(t)x(t) and y(t)y(t)y(t).

Flashcard 8: Identify dxdt\frac{dx}{dt}dtdx​ if x(t)=etx(t) = e^tx(t)=et.

Flashcard 9: What is dydx\frac{dy}{dx}dxdy​ for y(t)=t2y(t) = t^2y(t)=t2 and x(t)=t2x(t) = t^2x(t)=t2?

Flashcard 10: What is the derivative dydt\frac{dy}{dt}dtdy​ for y(t)=t2−5t+6y(t) = t^2 - 5t + 6y(t)=t2−5t+6?

Flashcard 11: What is the expression for dydx\frac{dy}{dx}dxdy​ if y(t)=t3y(t) = t^3y(t)=t3 and x(t)=tx(t) = tx(t)=t?

Flashcard 12: Find dxdt\frac{dx}{dt}dtdx​ if x(t)=t3−4t2x(t) = t^3 - 4t^2x(t)=t3−4t2.

Flashcard 13: State the second derivative formula in terms of ttt for x(t)x(t)x(t) and y(t)y(t)y(t).

Flashcard 14: Find dxdt\frac{dx}{dt}dtdx​ for x(t)=2t2+3tx(t) = 2t^2 + 3tx(t)=2t2+3t.

Flashcard 15: Find the expression for dydx\frac{dy}{dx}dxdy​ given y(t)=sin⁡(t)y(t) = \sin(t)y(t)=sin(t) and x(t)=cos⁡(t)x(t) = \cos(t)x(t)=cos(t).

Flashcard 16: Identify dydx\frac{dy}{dx}dxdy​ if y(t)=ety(t) = e^ty(t)=et and x(t)=e−tx(t) = e^{-t}x(t)=e−t.

Flashcard 17: What is dydx\frac{dy}{dx}dxdy​ for y(t)=tan⁡(t)y(t) = \tan(t)y(t)=tan(t) and x(t)=tx(t) = tx(t)=t?

Flashcard 18: State the expression for dydx\frac{dy}{dx}dxdy​ if y(t)=cos⁡(t)y(t) = \cos(t)y(t)=cos(t) and x(t)=sin⁡(t)x(t) = \sin(t)x(t)=sin(t).

Flashcard 19: Calculate dydx\frac{dy}{dx}dxdy​ when y(t)=t2+1y(t) = t^2 + 1y(t)=t2+1 and x(t)=3tx(t) = 3tx(t)=3t.

Flashcard 20: State the formula for the second derivative in parametric form.

Flashcard 21: Find dxdt\frac{dx}{dt}dtdx​ if x(t)=t3−4t2x(t) = t^3 - 4t^2x(t)=t3−4t2.

Flashcard 22: Identify dydx\frac{dy}{dx}dxdy​ for y(t)=t2+3ty(t) = t^2 + 3ty(t)=t2+3t and x(t)=2tx(t) = 2tx(t)=2t.

Flashcard 23: What is dydx\frac{dy}{dx}dxdy​ for y(t)=t2y(t) = t^2y(t)=t2 and x(t)=t2x(t) = t^2x(t)=t2?

Flashcard 24: What is the derivative dydt\frac{dy}{dt}dtdy​ for y(t)=t2−5t+6y(t) = t^2 - 5t + 6y(t)=t2−5t+6?

Flashcard 25: What is the expression for dydx\frac{dy}{dx}dxdy​ if y(t)=t3y(t) = t^3y(t)=t3 and x(t)=tx(t) = tx(t)=t?

Flashcard 26: What is dxdt\frac{dx}{dt}dtdx​ for x(t)=tan⁡(t)x(t) = \tan(t)x(t)=tan(t)?

Flashcard 27: What is dydx\frac{dy}{dx}dxdy​ for y(t)=ety(t) = e^ty(t)=et and x(t)=t2x(t) = t^2x(t)=t2?

Flashcard 28: State the second derivative formula in terms of ttt for x(t)x(t)x(t) and y(t)y(t)y(t).

Flashcard 29: Find dxdt\frac{dx}{dt}dtdx​ for x(t)=2t2+3tx(t) = 2t^2 + 3tx(t)=2t2+3t.

Flashcard 30: Find the expression for dydx\frac{dy}{dx}dxdy​ given y(t)=sin⁡(t)y(t) = \sin(t)y(t)=sin(t) and x(t)=cos⁡(t)x(t) = \cos(t)x(t)=cos(t).

Flashcard 1: What is the derivative $\frac{dy}{dt}$ for $y(t) = t^3 - 4t$ ?

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

Flashcard 3: What is $\frac{dy}{dx}$ for $y(t) = e^t$ and $x(t) = t^2$ ?

Flashcard 4: What is $\frac{dx}{dt}$ for $x(t) = \tan(t)$ ?

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

Flashcard 6: Identify $\frac{dy}{dx}$ for $y(t) = t^2 + 3t$ and $x(t) = 2t$ .

Flashcard 7: Identify the formula for $\frac{dy}{dx}$ in terms of $x(t)$ and $y(t)$ .

Flashcard 8: Identify $\frac{dx}{dt}$ if $x(t) = e^t$ .

Flashcard 9: What is $\frac{dy}{dx}$ for $y(t) = t^2$ and $x(t) = t^2$ ?

Flashcard 10: What is the derivative $\frac{dy}{dt}$ for $y(t) = t^2 - 5t + 6$ ?

Flashcard 11: What is the expression for $\frac{dy}{dx}$ if $y(t) = t^3$ and $x(t) = t$ ?

Flashcard 12: Find $\frac{dx}{dt}$ if $x(t) = t^3 - 4t^2$ .

Flashcard 13: State the second derivative formula in terms of $t$ for $x(t)$ and $y(t)$ .

Flashcard 14: Find $\frac{dx}{dt}$ for $x(t) = 2t^2 + 3t$ .

Flashcard 15: Find the expression for $\frac{dy}{dx}$ given $y(t) = \sin(t)$ and $x(t) = \cos(t)$ .

Flashcard 16: Identify $\frac{dy}{dx}$ if $y(t) = e^t$ and $x(t) = e^{-t}$ .

Flashcard 17: What is $\frac{dy}{dx}$ for $y(t) = \tan(t)$ and $x(t) = t$ ?

Flashcard 18: State the expression for $\frac{dy}{dx}$ if $y(t) = \cos(t)$ and $x(t) = \sin(t)$ .

Flashcard 19: Calculate $\frac{dy}{dx}$ when $y(t) = t^2 + 1$ and $x(t) = 3t$ .

Flashcard 21: Find $\frac{dx}{dt}$ if $x(t) = t^3 - 4t^2$ .

Flashcard 22: Identify $\frac{dy}{dx}$ for $y(t) = t^2 + 3t$ and $x(t) = 2t$ .

Flashcard 23: What is $\frac{dy}{dx}$ for $y(t) = t^2$ and $x(t) = t^2$ ?

Flashcard 24: What is the derivative $\frac{dy}{dt}$ for $y(t) = t^2 - 5t + 6$ ?

Flashcard 25: What is the expression for $\frac{dy}{dx}$ if $y(t) = t^3$ and $x(t) = t$ ?

Flashcard 26: What is $\frac{dx}{dt}$ for $x(t) = \tan(t)$ ?

Flashcard 27: What is $\frac{dy}{dx}$ for $y(t) = e^t$ and $x(t) = t^2$ ?

Flashcard 28: State the second derivative formula in terms of $t$ for $x(t)$ and $y(t)$ .

Flashcard 29: Find $\frac{dx}{dt}$ for $x(t) = 2t^2 + 3t$ .

Flashcard 30: Find the expression for $\frac{dy}{dx}$ given $y(t) = \sin(t)$ and $x(t) = \cos(t)$ .

Flashcard 1: What is the derivative $\frac{dy}{dt}$ for $y(t) = t^3 - 4t$ ?

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

Flashcard 3: What is $\frac{dy}{dx}$ for $y(t) = e^t$ and $x(t) = t^2$ ?

Flashcard 4: What is $\frac{dx}{dt}$ for $x(t) = \tan(t)$ ?

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

Flashcard 6: Identify $\frac{dy}{dx}$ for $y(t) = t^2 + 3t$ and $x(t) = 2t$ .

Flashcard 7: Identify the formula for $\frac{dy}{dx}$ in terms of $x(t)$ and $y(t)$ .

Flashcard 8: Identify $\frac{dx}{dt}$ if $x(t) = e^t$ .

Flashcard 9: What is $\frac{dy}{dx}$ for $y(t) = t^2$ and $x(t) = t^2$ ?

Flashcard 10: What is the derivative $\frac{dy}{dt}$ for $y(t) = t^2 - 5t + 6$ ?

Flashcard 11: What is the expression for $\frac{dy}{dx}$ if $y(t) = t^3$ and $x(t) = t$ ?

Flashcard 12: Find $\frac{dx}{dt}$ if $x(t) = t^3 - 4t^2$ .

Flashcard 13: State the second derivative formula in terms of $t$ for $x(t)$ and $y(t)$ .

Flashcard 14: Find $\frac{dx}{dt}$ for $x(t) = 2t^2 + 3t$ .

Flashcard 15: Find the expression for $\frac{dy}{dx}$ given $y(t) = \sin(t)$ and $x(t) = \cos(t)$ .

Flashcard 16: Identify $\frac{dy}{dx}$ if $y(t) = e^t$ and $x(t) = e^{-t}$ .

Flashcard 17: What is $\frac{dy}{dx}$ for $y(t) = \tan(t)$ and $x(t) = t$ ?

Flashcard 18: State the expression for $\frac{dy}{dx}$ if $y(t) = \cos(t)$ and $x(t) = \sin(t)$ .

Flashcard 19: Calculate $\frac{dy}{dx}$ when $y(t) = t^2 + 1$ and $x(t) = 3t$ .

Flashcard 21: Find $\frac{dx}{dt}$ if $x(t) = t^3 - 4t^2$ .

Flashcard 22: Identify $\frac{dy}{dx}$ for $y(t) = t^2 + 3t$ and $x(t) = 2t$ .

Flashcard 23: What is $\frac{dy}{dx}$ for $y(t) = t^2$ and $x(t) = t^2$ ?

Flashcard 24: What is the derivative $\frac{dy}{dt}$ for $y(t) = t^2 - 5t + 6$ ?

Flashcard 25: What is the expression for $\frac{dy}{dx}$ if $y(t) = t^3$ and $x(t) = t$ ?

Flashcard 26: What is $\frac{dx}{dt}$ for $x(t) = \tan(t)$ ?

Flashcard 27: What is $\frac{dy}{dx}$ for $y(t) = e^t$ and $x(t) = t^2$ ?

Flashcard 28: State the second derivative formula in terms of $t$ for $x(t)$ and $y(t)$ .

Flashcard 29: Find $\frac{dx}{dt}$ for $x(t) = 2t^2 + 3t$ .

Flashcard 30: Find the expression for $\frac{dy}{dx}$ given $y(t) = \sin(t)$ and $x(t) = \cos(t)$ .