University:
Amherst College
Course:
MATH 106 | Calculus with Elementary Functions
Academic year:

2022
Views:

290

Pages:

2
Author:

Davon Sims

Math 106, Spring 2022 Worksheet 2, Tuesday, February 22nd, 2022 Antiderivatives Definition: Take F, f functions defined on an interval I and suppose that F 0 (x) = f (x) on I. Then • F (x) is called AN Antiderivative of f (x) • F (x) + C is called the Most General Antiderivative of f (x), where C is any constant Real Number. Z We will use the notation f (x) dx to denote the Most General Antiderivative. x8 x7 dx = +C where +C represent all possible constants. 8 8 x d + C = x7 which looks correct to us. This means that dx 8 Z For example: Note that x8 x8 x8 √ x8 + 3 is an antiderivative of x7 . So is + 2022 as well as − 5 and + 3. 8 8 8 8 Hint: if you ever want to know whether you found the correct antiderivative, take the derivative of your answer and check that you return to the original function. Z 1. Write a general power rule for xn dx where n is any real number with n 6= −1. (We will learn the n = −1 case at the very end of this semester.) Compute each of the following Most General Antiderivatives Z 2. √ Z x dx 3. Z 6. 1 dx x9 Z 4. Z cos x dx 7. Z sin x dx 8. 1 √ dx x 2 sec x dx Z 5. 1 3 dx x7 Z 9. sec x tan x dx 3 3 1 4 1 3 4 For #10 − 11, consider f (x) = x + x 4 − 3 + + 4 + 4 − 3 4 3 x3 4x x x4 3 d 1 3 4 1 3 4 0 4 10. Compute f (x) = x+x − 3 + + 4 + 4 − 3 dx 4 3 x3 4x x x4 Z Z 3 3 1 4 1 3 4 11. Compute f (x) dx = x + x 4 − 3 + + 4 + 4 − 3 dx 4 3 x3 4x x x4 Find the Most General Antiderivative of the following functions. You may need to use some Algebra to prep the function pieces for each Power Rule and/or Sum or Difference Rule(s). Z 12. x3 (1 + x2 ) dx Z 13. √ x+7 x3 dx Z x2 + x(1 + x)2 dx Z −3 cos x − sec2 x − 7 sec x tan x − sin x dx 14. 15. x+ 16. Consider the curve y = 2x + sin x. Explain why the tangent lines of this curve are never horizontal. Find the indicated functions f (x) that also satisfies the given conditions: 17. f 0 (x) = 2 + sin x and f π 2 = 3. 18. f 0 (x) = x2 + 1 and f (1) = 3. 19. f 0 (x) = x(2 + √ x) and f (4) = 30. 1 20. f 00 (x) = √ + 3x2 and f 0 (1) = 2, f (1) = 0. x 21. Can you use a guess and check approach to compute the function f (x) where f 0 (x) = sin(3x)? Check your answer. How? Think about reversing the Chain Rule here... 22. CHALLENGE: Can you use a guess and check approach to compute the function f (x) sec x tan x where f 0 (x) = √ and f (0) = 7? Check your answer. sec x + 8 Turn in your own solutions. 2

Math 106, Spring 2022 Worksheet 2 Antiderivatives

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