Introduction to Related Rates - AP Calculus AB

1. Determine $\frac{dC}{dt}$ if $C = 2\text{π}r$ and $\frac{dr}{dt} = 5$ at $r = 4$.

   $\frac{dC}{dt} = 10\text{π}$. Circumference formula differentiated.

2. What is the derivative of $\frac{d}{dx}(\text{ln}(kx))$?

   $\frac{d}{dx}(\text{ln}(kx)) = \frac{1}{x}$. Chain rule cancels constant $k$.

3. What is the derivative of $\frac{d}{dx}(\text{ln}(x))$?

   $\frac{d}{dx}(\text{ln}(x)) = \frac{1}{x}$. Natural logarithm derivative formula.

4. Find $\frac{dV}{dt}$ if $V = \frac{4}{3}\times \pi r^3$ and $\frac{dr}{dt} = 2$ at $r = 3$.

   $\frac{dV}{dt} = 72\pi$. Volume formula for sphere, differentiated.

5. Determine $\frac{dC}{dt}$ for $C = 2\pi r$ if $r = 10$ and $\frac{dr}{dt} = 0.1$.

   $\frac{dC}{dt} = 0.2\pi$. Circumference: $\frac{dC}{dt} = 2\pi \cdot 0.1 = 0.2\pi$.

6. Determine $\frac{dA}{dt}$ for $A = s^2$ if $s = 10$ and $\frac{ds}{dt} = 2$.

   $\frac{dA}{dt} = 40$. Square area formula: $\frac{dA}{dt} = 2s \cdot 2 = 40$.

7. Identify the formula for the derivative of $\text{sin}(x)$.

   $\frac{d}{dx}(\text{sin}(x)) = \text{cos}(x)$. Sine derivative is cosine.

8. What is the relationship between rates of change in related rates problems?

   They are connected by the chain rule. Chain rule connects changing variables.

9. State the formula for the derivative of $\frac{d}{dx}(e^x)$.

   $\frac{d}{dx}(e^x) = e^x$. Exponential function derivative equals itself.

10. What is the derivative of $\frac{d}{dx}(\text{arctan}(x))$?

    $\frac{d}{dx}(\text{arctan}(x)) = \frac{1}{1+x^2}$. Inverse tangent derivative formula.

11. Calculate $\frac{dA}{dt}$ for $A = \pi r^2$ if $\frac{dr}{dt} = 1$ and $r = 5$.

    $\frac{dA}{dt} = 10\pi$. Circle area formula differentiated.

12. Determine $\frac{dy}{dt}$ for $y = x^2$ if $\frac{dx}{dt} = 4$ and $x = 3$.

    $\frac{dy}{dt} = 24$. Apply chain rule: $\frac{dy}{dt} = 2x \cdot 4 = 24$.

13. What is the derivative of $\frac{d}{dx}(\text{arccos}(x))$?

    $\frac{d}{dx}(\text{arccos}(x)) = -\frac{1}{\text{sqrt}(1-x^2)}$. Inverse cosine derivative is negative.

14. Calculate $\frac{dh}{dt}$ for $h = \text{tan}(t)$ when $t = \frac{\text{π}}{4}$.

    $\frac{dh}{dt} = 2$. At $t = \frac{\pi}{4}$, $\sec^2(\frac{\pi}{4}) = 2$.

15. What is the derivative of $\frac{d}{dx}(\text{e}^{kx})$?

    $\frac{d}{dx}(\text{e}^{kx}) = k\text{e}^{kx}$. Chain rule with exponential function.

16. What is the derivative of $\frac{d}{dx}(\text{log}_a{x})$?

    $\frac{d}{dx}(\text{log}_a{x}) = \frac{1}{x\text{ln}(a)}$. Logarithm derivative with base $a$.

17. What is the formula for the derivative of $\frac{f}{g}$?

    $(f/g)' = \frac{f'g - fg'}{g^2}$. Quotient rule for derivatives.

18. What is the chain rule for finding derivatives?

    $\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$. Composes derivatives for nested functions.

19. What is the formula for the derivative of $\frac{d}{dx}(\cos(x))$?

    $\frac{d}{dx}(\cos(x)) = -\sin(x)$. Cosine derivative is negative sine.

20. What is the derivative of $\frac{d}{dx}(a^x)$?

    $\frac{d}{dx}(a^x) = a^x\text{ln}(a)$. General exponential function derivative.

21. What is the derivative of $\frac{d}{dx}(\text{csc}(x))$?

    $\frac{d}{dx}(\text{csc}(x)) = -\text{csc}(x)\text{cot}(x)$. Cosecant derivative is negative.

22. What is the derivative of $\frac{d}{dx}(\text{cot}(x))$?

    $\frac{d}{dx}(\text{cot}(x)) = -\text{csc}^2(x)$. Cotangent derivative uses cosecant squared.

23. Which rule is primarily used in related rates problems?

    Chain rule. Links rates through composite functions.

24. What is the derivative of $\frac{d}{dx}(\text{sec}(x))$?

    $\frac{d}{dx}(\text{sec}(x)) = \text{sec}(x)\text{tan}(x)$. Secant derivative involves secant and tangent.

25. Find $\frac{dV}{dt}$ for $V = \frac{4}{3}\text{π}r^3$ if $r = 6$ and $\frac{dr}{dt} = 0.5$.

    $\frac{dV}{dt} = 72\text{π}$. Sphere volume: $\frac{dV}{dt} = 4\pi r^2 \cdot 0.5 = 72\pi$.

26. What is the derivative of $\frac{d}{dx}(\text{arcsin}(x))$?

    $\frac{d}{dx}(\text{arcsin}(x)) = \frac{1}{\text{sqrt}(1-x^2)}$. Inverse sine derivative formula.

27. Determine $\frac{dy}{dt}$ for $y = x^3$ if $x = 2$ and $\frac{dx}{dt} = 3$.

    $\frac{dy}{dt} = 36$. Power rule: $\frac{dy}{dt} = 3x^2 \cdot 3 = 36$.

28. What is the derivative of $\frac{d}{dx}(\frac{1}{x})$?

    $\frac{d}{dx}(\frac{1}{x}) = -\frac{1}{x^2}$. Power rule with $n = -1$.

29. Calculate $\frac{dA}{dt}$ for $A = \pi r^2$ if $r = 7$ and $\frac{dr}{dt} = 0.5$.

    $\frac{dA}{dt} = 7\pi$. Circle area: $\frac{dA}{dt} = 2\pi r \cdot 0.5 = 7\pi$.

30. What is the formula for the derivative of $\frac{d}{dx}(\text{cos}(x))$?

    $\frac{d}{dx}(\text{cos}(x)) = -\text{sin}(x)$. Cosine derivative is negative sine.

31. Find $\frac{dV}{dt}$ for $V = \frac{4}{3} \pi r^3$ if $r = 6$ and $\frac{dr}{dt} = 0.5$.

    $\frac{dV}{dt} = 72\pi$. Sphere volume: $\frac{dV}{dt} = 4\pi r^2 \cdot 0.5 = 72\pi$.

32. Identify the formula for the derivative of $x^n$.

    $\frac{d}{dx}(x^n) = nx^{n-1}$. Power rule for polynomial derivatives.

33. What is the basic strategy for solving related rates problems?

    Identify variables, relate them, differentiate, solve. Standard approach for related rates.

34. State the formula for the derivative of a product of two functions.

    $(f \times g)' = f' \times g + f \times g'$. Product rule for derivatives.

35. Determine $\frac{dC}{dt}$ for $C = 2\text{π}r$ if $r = 10$ and $\frac{dr}{dt} = 0.1$.

    $\frac{dC}{dt} = 0.2\text{π}$. Circumference: $\frac{dC}{dt} = 2\pi \cdot 0.1 = 0.2\pi$.

Study these flashcards