Estimating Derivatives of a Function - AP Calculus BC

1. Estimate $f'(3)$ for $f(x) = \sin x$ using $h = 0.01$ and the forward difference quotient.

   -0.9895. Forward formula: $\frac{\sin(3.01) - \sin(3)}{0.01} \approx -0.9895$

2. Identify the error: $f'(a) \approx \frac{f(a+h) + f(a-h)}{2h}$

   Correct: $f'(a) \approx \frac{f(a+h) - f(a-h)}{2h}$. Should subtract, not add, in the numerator.

3. Which difference quotient is more accurate: forward or symmetric?

   Symmetric. Averages left and right slopes for better approximation.

4. For $f(x) = x^3$, estimate $f'(2)$ using $h = 0.1$ and the symmetric difference quotient.

   12.01. Symmetric formula: $\frac{(2.1)^3 - (1.9)^3}{0.2} = 12.01$

5. Estimate $f'(0)$ for $f(x) = \sin(x^2)$ using $h = 0.01$ and the backward difference quotient.

   0.0000. Backward formula: derivative is 0 at $x=0$ for even functions.

6. Estimate $f'(3)$ for $f(x) = \cos(2x)$ using $h = 0.01$ and the backward difference quotient.

   -1.9159. Backward formula: $\frac{\cos(6) - \cos(5.98)}{0.01} \approx -1.9159$

7. Estimate $f'(3)$ for $f(x) = x^2 + x$ using $h = 0.1$ and the symmetric difference quotient.

   7.0. Symmetric formula gives exact derivative $2x+1=7$ at $x=3$.

8. Estimate $f'(5)$ for $f(x) = \log x$ using $h = 0.01$ and the backward difference quotient.

   0.1990. Backward formula: $\frac{\log(5) - \log(4.99)}{0.01} \approx 0.1990$

9. Estimate $f'(4)$ for $f(x) = \frac{1}{x^3}$ using $h = 0.1$ and the forward difference quotient.

   -0.0351. Forward formula: $\frac{1/(4.1)^3 - 1/64}{0.1} \approx -0.0351$

10. Estimate $f'(1)$ for $f(x) = \sin(2x)$ using $h = 0.001$ and the symmetric difference quotient.

    1.0806. Symmetric formula: $\frac{\sin(2.002) - \sin(1.998)}{0.002} \approx 1.0806$

11. Estimate $f'(4)$ for $f(x) = \frac{1}{x^2}$ using $h = 0.1$ and the backward difference quotient.

    -0.1251. Backward formula: $\frac{1/16 - 1/15.21}{0.1} \approx -0.1251$

12. Estimate $f'(3)$ for $f(x) = x^2 + 3x$ using $h = 0.1$ and the forward difference quotient.

    9.0. Forward formula gives exact derivative $2x+3=9$ at $x=3$.

13. Estimate $f'(0)$ for $f(x) = \cos x$ using $h = 0.01$ and the forward difference quotient.

    -0.0005. Forward formula: $\frac{\cos(0.01) - \cos(0)}{0.01} \approx -0.0005$

14. Estimate $f'(1)$ for $f(x) = \sqrt{x}$ using $h = 0.001$ and the backward difference quotient.

    0.4995. Backward formula: $\frac{\sqrt{1} - \sqrt{0.999}}{0.001} \approx 0.4995$

15. Estimate $f'(2)$ for $f(x) = \ln(1+x^2)$ using $h = 0.1$ and the symmetric difference quotient.

    0.8000. Symmetric formula: $\frac{\ln(5.41) - \ln(4.01)}{0.2} \approx 0.8000$

16. For $f(x) = x^4$, estimate $f'(1)$ using $h = 0.1$ and the forward difference quotient.

    4.0301. Forward formula: $\frac{(1.1)^4 - 1}{0.1} \approx 4.0301$

17. What is the impact of a smaller $h$ on the accuracy of a derivative estimate?

    Increases accuracy. Smaller step sizes give more precise estimates.

18. Estimate $f'(0)$ for $f(x) = e^{2x}$ using $h = 0.01$ and the symmetric difference quotient.

    2.0000. Symmetric formula: $\frac{e^{0.02} - e^{-0.02}}{0.02} \approx 2.0000$

19. Identify the error: $f'(a) \approx \frac{f(a+h) + f(a-h)}{2h}$

    Correct: $f'(a) \approx \frac{f(a+h) - f(a-h)}{2h}$. Should subtract, not add, in the numerator.

20. Estimate $f'(0)$ for $f(x) = \cos x$ using $h = 0.01$ and the forward difference quotient.

    -0.0005. Forward formula: $\frac{\cos(0.01) - \cos(0)}{0.01} \approx -0.0005$

21. Estimate $f'(2)$ for $f(x) = x^3 - 3x^2 + 2x$ using $h = 0.1$ and the forward difference quotient.

    -2.999. Forward formula for $f'(x) = 3x^2 - 6x + 2$ at $x=2$.

22. Estimate $f'(3)$ for $f(x) = \tan^{-1}(x)$ using $h = 0.01$ and the backward difference quotient.

    0.0995. Backward formula: $\frac{\tan^{-1}(3) - \tan^{-1}(2.99)}{0.01} \approx 0.0995$

23. Estimate $f'(2)$ for $f(x) = \sqrt{1+x}$ using $h = 0.1$ and the symmetric difference quotient.

    0.3536. Symmetric formula: $\frac{\sqrt{3.1} - \sqrt{2.9}}{0.2} \approx 0.3536$

24. Estimate $f'(2)$ for $f(x) = \ln x$ using $h = 0.01$ and the backward difference quotient.

    0.5001. Backward formula: $\frac{\ln(2) - \ln(1.99)}{0.01} \approx 0.5001$

25. Which value of $h$ gives a more accurate estimate: $h=0.1$ or $h=0.01$?

    $h=0.01$. Smaller $h$ values reduce approximation error.

26. Estimate $f'(5)$ for $f(x) = x^2$ using $h = 0.01$ and the forward difference quotient.

    10.01. Forward formula: $\frac{(5.01)^2 - 25}{0.01} = 10.01$

27. Estimate $f'(0)$ for $f(x) = \log(1+x)$ using $h = 0.01$ and the symmetric difference quotient.

    0.9950. Symmetric formula: $\frac{\log(1.01) - \log(0.99)}{0.02} \approx 0.9950$

28. Estimate $f'(1)$ for $f(x) = e^{-x}$ using $h = 0.001$ and the backward difference quotient.

    -0.368. Backward formula: $\frac{e^{-1} - e^{-1.001}}{0.001} \approx -0.368$

29. Estimate $f'(0)$ for $f(x) = \ln(1+x)$ using $h = 0.1$ and the symmetric difference quotient.

    0.9950. Symmetric formula: $\frac{\ln(1.1) - \ln(0.9)}{0.2} \approx 0.9950$

30. What is the formula for the backward difference quotient to estimate the derivative at a point?

    $f'(a) \approx \frac{f(a) - f(a-h)}{h}$. Uses the point before $a$ to estimate the slope.

31. Estimate $f'(0)$ for $f(x) = \frac{1}{1+x}$ using $h = 0.1$ and the symmetric difference quotient.

    -0.9990. Symmetric formula: $\frac{1/1.1 - 1/0.9}{0.2} \approx -0.9990$

32. Estimate $f'(2)$ for $f(x) = \tan x$ using $h = 0.01$ and the forward difference quotient.

    2.0199. Forward formula: $\frac{\tan(2.01) - \tan(2)}{0.01} \approx 2.0199$

33. Estimate $f'(1)$ for $f(x) = 2^x$ using $h = 0.001$ and the symmetric difference quotient.

    1.3863. Symmetric formula: $\frac{2^{1.001} - 2^{0.999}}{0.002} \approx 1.3863$

34. Which method reduces error due to truncation: forward or symmetric difference quotient?

    Symmetric. Balances left and right approximation errors.

35. What is the formula for the forward difference quotient to estimate the derivative at a point?

    $f'(a) \approx \frac{f(a+h) - f(a)}{h}. Uses the point after $a$ to estimate the slope.

36. Estimate $f'(1)$ for $f(x) = x \cdot e^x$ using $h = 0.001$ and the forward difference quotient.

    5.4379. Forward formula: $\frac{1.001 \cdot e^{1.001} - e}{0.001} \approx 5.4379$

37. Estimate $f'(2)$ for $f(x) = x^2$ using $h = 0.01$ and the symmetric difference quotient.

    4.0001. Symmetric formula: $\frac{(2.01)^2 - (1.99)^2}{0.02} = 4.0001$

38. Estimate $f'(4)$ for $f(x) = \frac{1}{x}$ using $h = 0.1$ and the forward difference quotient.

    -0.0625. Forward formula: $\frac{1/4.1 - 1/4}{0.1} \approx -0.0625$

39. Estimate $f'(1)$ for $f(x) = e^x$ using $h = 0.001$ and the backward difference quotient.

    2.718. Backward formula: $\frac{e^1 - e^{0.999}}{0.001} \approx 2.718$

40. What is the formula to estimate the derivative of a function at a point using the symmetric difference quotient?

    $f'(a) \approx \frac{f(a+h) - f(a-h)}{2h}$. Uses points on both sides of $a$ for better accuracy.

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