AP Calc AB Sample Problem Help (Integration)

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SUMMARY

The discussion focuses on solving the AP Calculus AB problem involving the function defined by its derivative f'(x) = sin(πe^x/2) with the initial condition f(0) = 1. The correct answer for f(2) is determined to be 1.157, achieved through numerical integration using the Nint feature on the TI-89 calculator. Additionally, a local linear approximation method is suggested, yielding a similar result. The participants emphasize the importance of using the Nint function effectively to compute definite integrals.

PREREQUISITES
  • Understanding of calculus concepts, specifically integration and derivatives.
  • Familiarity with the TI-89 calculator and its Nint feature.
  • Knowledge of numerical integration techniques.
  • Ability to perform local linear approximations in calculus.
NEXT STEPS
  • Explore the TI-89 calculator's Nint feature in detail.
  • Study numerical integration methods in calculus.
  • Learn about local linear approximation techniques and their applications.
  • Practice solving AP Calculus AB problems involving integration and derivatives.
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Students preparing for the AP Calculus AB exam, educators teaching calculus concepts, and anyone seeking to improve their skills in numerical integration and approximation methods.

Inspector Gadget
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This is on the sample multiple choice questions in the section in which calculators are permitted. I tried it on my TI-89, and it didn't give an answer...just returned what was input with the integral sign and such...

Given that...

f'(x) = sin(\frac{\pi \times e^x}{2})

... and f(0) = 1, find f(2).

Can someone help?
 
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Forgot...

A.) -1.819
B.) -0.843
C.) -0.819
D.) 0.157
E.) 1.157

Answer: E
 
Try Numerical Integration... Nint feature that is on the TI 89...

Press f3, alpha, b, OR scroll up until you hit the bottom on the list and find Nint.

Should look like...

Nint((sin(pi*e^x/2),x,0,2) then add f(0) to get f(2)... 1.157
 
You could also try local linear approximation:

f(x)\sim f(a)+f^\prime (x)\cdot (x-a)

which in this case would boil down to computing

1+2\sin ({{e^2 \pi}\over {2}} )

since your possible answers are all fairly different (hindsight: the correct answer is very different from the others), this estimation should give you a good enough idea to answer the question.
 

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