Average Value of Functions on Intervals - AP Calculus BC

1. Identify the average value of $f(x) = \cos^2(x)$ on $[0, \pi]$.

   $\frac{1}{2}$. Squared cosine uses identity $\cos^2 x = \frac{1 + \cos(2x)}{2}$.

2. Determine the average value of $f(x) = x^2 + 1$ on $[0, 3]$.

   $4$. Quadratic plus constant function integrated over $[0,3]$.

3. What is the average value of $f(x) = x^2 - x$ on $[0, 3]$?

   $2$. Quadratic minus linear term integrated over $[0,3]$.

4. Calculate the average value of $f(x) = \text{sin}(2x)$ on $[0, \frac{\pi}{2}]$.

   $\frac{2}{\pi}$. Double angle sine function integrated over quarter period.

5. What is the average value of $f(x) = x^4$ on $[0, 2]$?

   $\frac{16}{5}$. Fourth power function $x^4$ integrated from $0$ to $2$.

6. What is the average value of $f(x) = \text{e}^{2x}$ on $[0, 1]$?

   $\frac{\text{e}^2 - 1}{2}$. Double exponential function $e^{2x}$ integrated over unit interval.

7. Identify the average value of $f(x) = \text{cos}^2(x)$ on $[0, \text{pi}]$.

   $\frac{1}{2}$. Squared cosine uses identity $\cos^2 x = \frac{1 + \cos(2x)}{2}$.

8. What is the average value of $f(x) = \frac{1}{x}$ on $[1, \text{e}]$?

   $1$. Natural logarithm of $x$ gives $\ln(e) - \ln(1) = 1$.

9. Find the average value of $f(x) = 3x + 1$ on $[1, 4]$.

   $7$. Integrate linear function $3x + 1$ over $[1,4]$ and divide by $3$.

10. Identify the average value of $f(x) = \sin x$ from $0$ to $\frac{\pi}{2}$.

    $\frac{2}{\pi}$. Integrate $\sin x$ from $0$ to $\frac{\pi}{2}$ and divide by $\frac{\pi}{2}$.

11. Calculate the average value of $f(x) = \cos x$ on $[0, \pi]$.

    $0$. Cosine function completes half cycle from $0$ to $\pi$.

12. Which integral represents the average value of $f(x)$ on $[a, b]$?

    $\frac{1}{b-a} \times \int_a^b f(x) \, dx$. The fundamental formula for computing average value of any function.

13. What is the average value of $f(x) = 8 - x^2$ on $[-2, 2]$?

    $6$. Symmetric function $8 - x^2$ on symmetric interval $[-2,2]$.

14. Find the average value of $f(x) = 4x^3$ on $[0, 1]$.

    $1$. Integrate $4x^3$ from $0$ to $1$ and divide by interval length.

15. For $f(x) = x$, find the average value on $[1, 4]$.

    $2.5$. Linear function $x$ has average equal to midpoint of interval.

16. What is the average value of $f(x) = x^4 - 2x^2$ on $[-1, 1]$?

    $-\frac{2}{5}$. Even function with negative quadratic term on symmetric interval.

17. Calculate the average value of $f(x) = \text{cos}x$ on $[0, \text{pi}]$.

    $0$. Cosine function completes half cycle from $0$ to $\pi$.

18. For $f(x) = 5$, what is the average value on $[2, 5]$?

    $5$. Constant functions have average value equal to the constant.

19. Calculate the average value of $f(x) = \frac{1}{x}$ on $[1, 4]$.

    $\frac{\ln(4)}{3}$. Integrate $\frac{1}{x}$ to get $\ln(x)$, evaluate from $1$ to $4$.

20. Identify the average value of $f(x) = \text{e}^x$ on $[1, 3]$.

    $\frac{\text{e}^3 - \text{e}}{2}$. Exponential function over interval $[1,3]$ of length $2$.

21. For $f(x) = \text{sin}^2(x)$, what is the average value on $[0, \text{pi}]$?

    $\frac{1}{2}$. Squared sine uses identity $\sin^2 x = \frac{1 - \cos(2x)}{2}$.

22. What is the average value of $f(x) = 2x$ on $[0, 3]$?

    $3$. Linear function $2x$ integrated from $0$ to $3$ gives average $3$.

23. Calculate the average value of $f(x) = \text{e}^{-x}$ on $[0, 1]$.

    $1 - \frac{1}{\text{e}}$. Negative exponential function integrated over unit interval.

24. Determine the average value of $f(x) = \text{tan}(x)$ on $[0, \frac{\text{pi}}{4}]$.

    $\text{ln}(2)$. Tangent function integrated gives $-\ln(\cos x)$ from $0$ to $\frac{\pi}{4}$.

25. What is the average value of $f(x) = \frac{1}{x^2}$ on $[1, 2]$?

    $\frac{3}{4}$. Negative power function $x^{-2}$ integrated from $1$ to $2$.

26. What is the average value of $f(x) = 3x^2 - 2x$ on $[0, 1]$?

    $\frac{4}{3}$. Cubic minus quadratic function integrated over unit interval.

27. Which theorem relates to the average value of continuous functions?

    Mean Value Theorem for Integrals. Guarantees existence of a point where function equals its average value.

28. Find the average value of $f(x) = \frac{1}{x}$ on $[2, 4]$.

    $\frac{\text{ln}(2)}{2}$. Reciprocal function integrated over interval $[2,4]$ of length $2$.

29. What is the average value of $f(x) = x^3$ on $[-1, 1]$?

    $0$. Odd function $x^3$ on symmetric interval has zero average.

30. What is the average value of $f(x) = x^2$ on $[0, 2]$?

    $\frac{2}{3}$. Integrate $x^2$ from $0$ to $2$, then divide by interval length $2$.

31. What is the average value of $f(x) = x^3 + x^2$ on $[0, 1]$?

    $\frac{3}{4}$. Sum of cubic and quadratic terms integrated over unit interval.

32. What does the average value formula calculate?

    The mean height of the function over an interval. Represents the constant height that gives same area as function.

33. Find the average value of $f(x) = 2x^2 + 3x$ on $[1, 3]$.

    $14$. Quadratic plus linear function integrated over $[1,3]$.

34. Identify the average value of $f(x) = \text{sin}x$ from $0$ to $\frac{\text{pi}}{2}$.

    $\frac{2}{\text{pi}}$. Integrate $\sin x$ from $0$ to $\frac{\pi}{2}$ and divide by $\frac{\pi}{2}$.

35. Identify the average value of $f(x) = 6x^2$ on $[1, 2]$.

    $14$. Quadratic function $6x^2$ integrated over unit interval $[1,2]$.

36. Find the average value of $f(x) = \text{ln}(x)$ on $[1, \text{e}]$.

    $1 - \frac{1}{\text{e}}$. Natural logarithm integrated by parts over $[1,e]$.

37. Calculate the average value of $f(x) = \sin(2x)$ on $[0, \frac{\pi}{2}]$.

    $\frac{2}{\pi}$. Double angle sine function integrated over quarter period.

38. Calculate the average value of $f(x) = \text{cos}(2x)$ on $[0, \frac{\text{pi}}{2}]$.

    $0$. Double angle cosine completes full cycle over given interval.

39. Determine the average value of $f(x) = \frac{1}{x^3}$ on $[1, 2]$.

    $\frac{7}{24}$. Cubic negative power function integrated from $1$ to $2$.

40. Find the average value of $f(x) = \ln(x)$ on $[1, e]$

    $1 - \frac{1}{e}$. Natural logarithm integrated by parts over $[1,e]$

41. What is the average value of $f(x) = \text{e}^x$ on $[0, 2]$?

    $\frac{\text{e}^2 - 1}{2}$. Exponential function over interval of length $2$.

42. For $f(x) = e^x$, what is the average value on $[0, 1]$?

    $\frac{e - 1}{1}$. Exponential function integrated gives $e - 1$ over unit interval.

43. What is the average value of $f(x) = \text{ln}(x)$ on $[1, 2]$?

    $\text{ln}(2) - \frac{1}{2}$. Logarithm integrated by parts over interval $[1,2]$.

44. State the formula for the average value of a function $f(x)$ on $[a, b]$.

    $\frac{1}{b-a} \times \int_a^b f(x) \, dx$. Uses the integral divided by interval length to find mean height.

45. What is the average value of $f(x) = e^x$ on $[0, 2]$?

    $\frac{e^2 - 1}{2}$. Exponential function over interval of length $2$.

46. Identify the average value of $f(x) = e^x$ on $[1, 3]$.

    $\frac{e^3 - e}{2}$. Exponential function over interval $[1,3]$ of length $2$.

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