Rate of Change at a Point - AP Calculus AB

1. How do you symbolically represent the derivative of $f(x)$ at $x=a$?

   $f'(a)$. Standard notation for the derivative at a specific point.

2. Which mathematical concept represents the instantaneous rate of change?

   Derivative. The derivative measures instantaneous rate of change.

3. Find the derivative of $f(x)=\text{cos}(x)$.

   $-\text{sin}(x)$. The derivative of cosine is negative sine.

4. Calculate $f'(x)$ for $f(x)=\text{ln}(x)$.

   $\frac{1}{x}$. The derivative of the natural logarithm function.

5. If $f(x) = x^2 + 3x$, what is $f'(x)$?

   $2x + 3$. Apply power rule to each term.

6. Find the instantaneous rate of change for $f(x)=x^2+2x$ at $x=3$.

   8. $f'(x) = 2x + 2$, so $f'(3) = 8$.

7. How is the instantaneous rate of change at a point $x=a$ defined?

   Derivative $f'(a)$. The derivative gives the instantaneous rate of change at a specific point.

8. Identify the geometric representation of $f'(a)$.

   Slope of tangent line at $x=a$. The derivative equals the slope of the line tangent to the curve.

9. Identify the derivative of $f(x)=\text{e}^x$ at $x=0$.

   1. Since $f'(x) = e^x$ and $e^0 = 1$.

10. What is the relationship between average rate of change and secant lines?

    Average rate = Slope of secant line. Secant lines connect two points, giving average rate.

11. What is the derivative of the constant function $f(x)=c$?

    0. Constants have no rate of change.

12. Find $f'(a)$ for $f(x) = \frac{1}{x}$ at $x=2$.

    $-\frac{1}{4}$. Using $f'(x) = -\frac{1}{x^2}$, so $f'(2) = -\frac{1}{4}$.

13. What is the formula for average rate of change of $f(x)$ from $x=a$ to $x=b$?

    $\frac{f(b) - f(a)}{b - a}$. Standard formula for average rate of change over an interval.

14. In which scenario does average rate of change equal instantaneous rate of change?

    When $f(x)$ is linear. Linear functions have constant slope everywhere.

15. What does the average rate of change of a function represent?

    Slope of secant line. Connects two points on the function graph.

16. What is the instantaneous rate of change of $f(x)=5$ at any point?

    $0$. Constant functions have zero derivative.

17. Which function has a constant average rate of change?

    Linear function. Linear functions have the same rate everywhere.

18. What is the derivative of $f(x)=\tan(x)$?

    $\sec^2(x)$. The derivative of tangent is secant squared.

19. What does the instantaneous rate of change of a function represent?

    Slope of tangent line. The tangent touches the curve at exactly one point.

20. What is the geometric interpretation of the derivative?

    Slope of the tangent line. The derivative represents the slope of the tangent line.

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

    4. $\frac{9-1}{3-1} = \frac{8}{2} = 4$

22. What is the interpretation of $f'(x)$ in terms of velocity?

    Velocity function. When position is $f(x)$, $f'(x)$ represents velocity.

23. Compute the instantaneous rate of change for $f(x)=x^3$ at $x=1$.

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

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

    7. $\frac{8-1}{2-1} = 7$

25. What is $f'(x)$ for $f(x)=e^x$?

    $e^x$. The exponential function is its own derivative.

26. Find the instantaneous rate of change of $f(x)=x^2$ at $x=2$.

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

27. Determine $f'(3)$ for $f(x)=3x^2 + 2x$.

    20. $f'(x) = 6x + 2$, so $f'(3) = 18 + 2 = 20$.

28. Calculate the average rate of change of $f(x)=3x+5$ from $x=2$ to $x=5$.

    3. Linear functions have constant slope of 3.

29. What is the derivative of $x^n$ using power rule?

    $nx^{n-1}$. The power rule for differentiation.

30. State the formal definition of a derivative at a point $x=a$.

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

31. Determine $f'(x)$ if $f(x)=\frac{1}{x}$.

    $-\frac{1}{x^2}$. Derivative of $x^{-1}$ using the power rule.

32. What is the derivative of a constant function?

    Zero. Constants have no rate of change.

33. If $f(x)=\text{sin}(x)$, what is $f'(x)$?

    $\text{cos}(x)$. The derivative of sine is cosine.

34. Identify the formula for the difference quotient.

    $\frac{f(a+h) - f(a)}{h}$. The ratio used in the limit definition of derivatives.

35. Determine the derivative of $f(x) = \text{ln}(x)$ at $x=1$.

    1. Using $f'(x) = \frac{1}{x}$, so $f'(1) = 1$.

36. Find the derivative of $f(x)=\cos(x)$.

    $-\sin(x)$. The derivative of cosine is negative sine.

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