Rate of Change at a Point - AP Calculus BC

1. What is the derivative of $f(x) = x^3$ at $x = 2$?

   $12$. Using power rule: $f'(x) = 3x^2$, so $f'(2) = 12$.

2. What is the instantaneous rate of change of $f(x) = \text{ln}(x)$ at $x = 1$?

   $1$. The derivative of $\ln(x)$ is $\frac{1}{x}$.

3. Find the average rate of change of $f(x) = x^2$ from $x = 1$ to $x = 3$.

   $4$. Using $\frac{f(3)-f(1)}{3-1} = \frac{9-1}{2} = 4$.

4. Identify $f'(x)$ for $f(x) = x^2 + 2x + 1$ using the power rule.

   $2x + 2$. Applying power rule to each term separately.

5. Find $f'(x)$ for $f(x) = \text{e}^x + \text{e}^{-x}$ using differentiation.

   $\text{e}^x - \text{e}^{-x}$. Differentiating each exponential term separately.

6. Calculate the average rate of change of $f(x) = \text{cos}(x)$ over $[0, \text{π}]$.

   $-\frac{2}{\text{π}}$. Using $\frac{\cos(\pi) - \cos(0)}{\pi - 0} = \frac{-1-1}{\pi} = -\frac{2}{\pi}$.

7. What is the instantaneous rate of change of $f(x) = 2x^2 - x$ at $x = 2$?

   $7$. Using $f'(x) = 4x - 1$, so $f'(2) = 7$.

8. Identify the instantaneous rate of change of $f(x) = 3x$ at $x = 2$.

   $3$. The derivative of $3x$ is constant: $3$.

9. What is the derivative of $f(x) = \text{e}^{2x}$ at $x = 0$?

   $2$. Using chain rule: $\frac{d}{dx}[e^{2x}] = 2e^{2x}$.

10. Find the derivative of $f(x) = \text{ln}(x^2)$ at $x = 1$.

    $2$. Using chain rule: $\frac{d}{dx}[\ln(x^2)] = \frac{2x}{x^2} = \frac{2}{x}$.

11. What is the instantaneous rate of change of $f(x) = x^3 + x$ at $x = 1$?

    $4$. Using $f'(x) = 3x^2 + 1$, so $f'(1) = 4$.

12. What is the instantaneous rate of change of $f(x) = x^2 - 4x$ at $x = 3$?

    $2$. Using $f'(x) = 2x - 4$, so $f'(3) = 2$.

13. Determine the instantaneous rate of change of $f(x) = x^2 - 3x$ at $x = 1$.

    $-1$. Using $f'(x) = 2x - 3$, so $f'(1) = -1$.

14. What is the derivative of $f(x) = \text{cos}(2x)$ at $x = \frac{\text{π}}{4}$?

    $-2\text{sin}(2x)$. Using chain rule on $\cos(2x)$ gives $-2\sin(2x)$.

15. Find $f'(x)$ for $f(x) = 5x^2$ using differentiation.

    $10x$. Using power rule: $\frac{d}{dx}[5x^2] = 10x$.

16. What is the average rate of change of $f(x) = x^2 + 4x$ over $[2, 5]$?

    $11$. Using $\frac{(25+20)-(4+8)}{5-2} = \frac{45-12}{3} = 11$.

17. Find the average rate of change of $f(x) = \text{e}^x$ over $[0, 1]$.

    $\text{e} - 1$. Using $\frac{e^1 - e^0}{1 - 0} = e - 1$.

18. What is the rate of change of $f(x) = x^3$ from $x = 2$ to $x = 4$?

    $28$. Using $\frac{f(4) - f(2)}{4 - 2} = \frac{64 - 8}{2} = 28$.

19. State the derivative definition for instantaneous rate of change at $x = a$.

    $f'(a) = \frac{d}{dx}f(x)|_{x=a}$. The derivative evaluated at the specific point.

20. What is the slope of the tangent line to $f(x) = x^2$ at $x = 1$?

    $2$. Using power rule: $f'(x) = 2x$, so $f'(1) = 2$.

21. Determine $f'(x)$ for $f(x) = 7$ using basic derivative rules.

    $0$. The derivative of any constant is zero.

22. Calculate the average rate of change of $f(x) = \cos(x)$ over $[0, \pi]$.

    $-\frac{2}{\pi}$. Using $\frac{\cos(\pi) - \cos(0)}{\pi - 0} = \frac{-1-1}{\pi} = -\frac{2}{\pi}$.

23. Calculate the average rate of change of $f(x) = 2x + 3$ from $x = 1$ to $x = 4$.

    $2$. Linear functions have constant rate of change equal to slope.

24. Calculate the derivative of $f(x) = \text{sin}(2x)$ at $x = 0$.

    $2$. Using chain rule: $\frac{d}{dx}[\sin(2x)] = 2\cos(2x)$.

25. What is the slope of the tangent line to $f(x) = x^3$ at $x = -1$?

    $3$. Using $f'(x) = 3x^2$, so $f'(-1) = 3$.

26. What is the derivative of $f(x) = \text{e}^x \text{sin}(x)$ at $x = 0$?

    $1$. Using product rule: $(e^x \sin x)' = e^x(\sin x + \cos x)$.

27. State the derivative of $f(x) = \frac{1}{x}$ at $x = 1$.

    $-1$. The derivative of $\frac{1}{x}$ is $-\frac{1}{x^2}$.

28. What is the formula to find the derivative of $f(x)$ using limits?

    $f'(x) = \text{lim}_{h \to 0} \frac{f(x+h) - f(x)}{h}$. The limit definition of derivative using difference quotient.

29. What is the geometric interpretation of the instantaneous rate of change?

    Slope of the tangent line at $x = a$. Tangent line touches the curve at exactly one point.

30. What is the derivative of $f(x) = \text{cos}(x)$ at $x = 0$?

    $0$. The derivative of $\cos(x)$ is $-\sin(x)$, and $\sin(0) = 0$.

31. What is the derivative of $f(x) = \text{sin}(x)$ at $x = \frac{\text{π}}{2}$?

    $0$. The derivative of $\sin(x)$ is $\cos(x)$, and $\cos(\frac{\pi}{2}) = 0$.

32. What is the formula for the average rate of change of $f(x)$ over $[a, b]$?

    $\frac{f(b) - f(a)}{b - a}$. Standard formula: change in function divided by change in input.

33. Identify the instantaneous rate of change of $f(x) = x^4$ at $x = 1$.

    $4$. Using power rule: $f'(x) = 4x^3$, so $f'(1) = 4$.

34. Find the derivative of $f(x) = e^x$ at $x = 0$.

    $1$. The derivative of $e^x$ is $e^x$, and $e^0 = 1$.

35. Determine the instantaneous rate of change of $f(x) = 3x^2 + 4$ at $x = 3$.

    $18$. Using $f'(x) = 6x$, so $f'(3) = 18$.

36. What is the geometric interpretation of the average rate of change?

    Slope of the secant line over $[a, b]$. Secant line connects two points on the curve.

37. Calculate the derivative of $f(x) = \text{tan}(x)$ at $x = 0$.

    $1$. The derivative of $\tan(x)$ is $\sec^2(x) = 1$ at $x = 0$.

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