Linear and Quadratic Rates of Change - AP Precalculus

1. What is the instantaneous rate of change of $f(x) = 2x^2$ at $x = -1$?

   -4. Find $f'(x) = 4x$, then evaluate at $x = -1$.

2. State the formula for finding the slope of a secant line over $[a, b]$.

   $\frac{f(b) - f(a)}{b - a}$. Same formula as average rate of change between two points.

3. State the formula for the derivative of $f(x) = ax^2 + bx + c$.

   $2ax + b$. General form of derivative for quadratic functions.

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

   3. Find $f'(x) = 3x^2$, then evaluate at $x = 1$.

5. Find the average rate of change of $f(x) = x^2 + 2x - 1$ over $[2, 4]$.

   8. Calculate $\frac{f(4) - f(2)}{4 - 2} = \frac{23 - 7}{2} = 8$.

6. State the formula for finding the slope of a secant line over $ [a, b] $.

   $\frac{f(b) - f(a)}{b - a}$. Same formula as average rate of change between two points.

7. Find the slope of the tangent line to $f(x) = x^2 - 5x + 6$ at $x = 3$.

   1. Find $f'(x) = 2x - 5$, then substitute $x = 3$.

8. Find the slope of the tangent line to $f(x) = 4x^2 - x$ at $x = 1$.

   7. Find $f'(x) = 8x - 1$, then evaluate at $x = 1$.

9. Identify the expression for the derivative of $f(x) = x^3 - 2x$.

   $3x^2 - 2$. Apply power rule to each term: $(x^3)' = 3x^2$ and $(2x)' = 2$.

10. State the formula for the derivative of $f(x) = 3x^2 + 4x + 5$.

    $6x + 4$. Differentiate each term using power rule and sum rule.

11. Find the average rate of change of $f(x) = x^2 + 4x$ over $[3, 6]$.

    13. Calculate $\frac{f(6) - f(3)}{6 - 3} = \frac{60 - 21}{3} = 13$.

12. Identify the derivative of $f(x) = 2x^3 - x^2 + 4$.

    $6x^2 - 2x$. Apply power rule: $(2x^3)' = 6x^2$ and $(-x^2)' = -2x$.

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

    0. Find $f'(x) = 3x^2 - 6x$, then substitute $x = 2$.

14. State the derivative of $f(x) = x^2 + 5x - 3$.

    $2x + 5$. Differentiate each term using power rule and constant rule.

15. Find the slope of the tangent line to $f(x) = x^2 + 3x$ at $x = 2$.

    7. Find $f'(x) = 2x + 3$, then substitute $x = 2$.

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

    $6x + 2$. Apply power rule: $(x^n)' = nx^{n-1}$ and sum rule.

17. What is the derivative of $f(x) = -x^2 + 3x$?

    $-2x + 3$. Apply power rule and sum rule to find the derivative.

18. Identify the average rate of change of $f(x) = 3x - 4$ over $[2, 5]$.

    3. Linear functions have constant rate equal to the coefficient of $x$.

19. Identify the average rate of change of $f(x) = 2x + 3$ over $[1, 4]$.

    2. Linear functions have constant slope, which equals the coefficient of $x$.

20. Identify the average rate of change of $f(x) = -x^2 + 4$ over $[0, 2]$.

    -2. Calculate $\frac{f(2) - f(0)}{2 - 0} = \frac{0 - 4}{2} = -2$.

21. State the relationship between secant and tangent lines in context of rates of change.

    Secant: average rate, Tangent: instantaneous rate. Secant connects two points; tangent touches at one point.

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

    6. Find $f'(x) = 2x$, then evaluate at $x = 3$.

23. Identify the slope of the tangent line to $f(x) = x^3 - 3x$ at $x = 1$.

    0. Find $f'(x) = 3x^2 - 3$, then evaluate at $x = 1$.

24. What is the average rate of change of $f(x) = 4x - 7$ over $[0, 2]$?

    $4$. Linear functions have constant rate of change equal to the slope.

25. What is the instantaneous rate of change of $f(x) = x^2 + 2x$ at $x = 0$?

    2. Find $f'(x) = 2x + 2$, then substitute $x = 0$.

26. What is the derivative of $f(x) = x^2 - 4x + 2$?

    $2x - 4$. Apply power rule and linear term differentiation.

27. What is the instantaneous rate of change of $f(x) = 3x^2 - 5x$ at $x = 4$?

    19. Find $f'(x) = 6x - 5$, then evaluate at $x = 4$.

28. Find the average rate of change of $f(x) = 5x^2 + 2x$ over $[1, 3]$.

    22. Calculate $\frac{f(3) - f(1)}{3 - 1} = \frac{51 - 7}{2} = 22$.

29. What is the derivative of $f(x) = 5x^2 - 3x + 1$?

    $10x - 3$. Differentiate using power rule on each term.

30. What is the instantaneous rate of change of $f(x) = x^2 - x$ at $x = 0$?

    -1. Find $f'(x) = 2x - 1$, then substitute $x = 0$.

31. Identify the average rate of change of $f(x) = 6x - x^2$ over $[0, 3]$.

    3. Calculate $\frac{f(3) - f(0)}{3 - 0} = \frac{9 - 0}{3} = 3$.

32. What is the average rate of change of $f(x) = -x^2 + 2x$ over $[1, 5]$?

    -4. Calculate $\frac{f(5) - f(1)}{5 - 1} = \frac{-15 - 1}{4} = -4$.

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

    $\frac{f(b) - f(a)}{b - a}$. This is the slope formula for secant lines connecting two points.

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

    16. Find $f'(x) = 8x$, then evaluate at $x = 2$.

35. What is the derivative of $f(x) = 7x^2$?

    $14x$. Use power rule on $7x^2$ to get $7 \cdot 2x = 14x$.

36. State the formula for the instantaneous rate of change of $f(x)$ at $x = a$.

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

37. What is the average rate of change of $f(x) = x^2$ over $[1, 3]$?

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

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