Chapter 4 | Applications of Derivatives
497
4.10 EXERCISES For the following exercises, show that F ( x ) are antiderivatives of f ( x ). 465. F ( x ) =5 x 3 +2 x 2 +3 x +1, f ( x ) =15 x 2 +4 x +3 466. F ( x ) = x 2 +4 x +1, f ( x ) =2 x +4 467. F ( x ) = x 2 e x , f ( x ) = e x ⎛ ⎝ x 2 +2 x ⎞ ⎠
483. f ( x ) = sin 2 ( x )cos( x ) 484. f ( x ) =0 2 ( x )+ 1 x 2 486. f ( x ) =csc x cot x +3 x 485. f ( x ) = 1 2 csc
487. f ( x ) =4csc x cot x −sec x tan x 488. f ( x ) =8sec x (sec x −4tan x )
468. F ( x ) =cos x , f ( x ) =−sin x 469. F ( x ) = e x , f ( x ) = e x
e −4 x +sin x
489. f ( x ) = 1 2
For the following exercises, find the antiderivative of the function. 470. f ( x ) = 1 x 2 + x 471. f ( x ) = e x −3 x 2 +sin x 472. f ( x ) = e x +3 x − x 2 473. f ( x ) = x −1+4sin(2 x ) For the following exercises, find the antiderivative F ( x ) of each function f ( x ).
For the following exercises, evaluate the integral. 490. ∫ (−1) dx 491. ∫ sin xdx 492. ∫ (4 x + x ) dx 493. ∫ 3 x 2 +2 x 2 dx 494. ∫ (sec x tan x +4 x ) dx 495. ∫ ⎛ ⎝ 4 x + x 4 ⎞ ⎠ dx 496. ∫ ⎛ ⎝ x −1/3 − x 2/3 ⎞ ⎠ dx 497. ∫ 14 x 3 +2 x +1 x 3 dx 498. ∫ ( e x + e − x ) dx For the following exercises, solve the initial value problem. 499. f ′( x ) = x −3 , f (1) =1
474. f ( x ) =5 x 4 +4 x 5 475. f ( x ) = x +12 x 2 476. f ( x ) = 1 x 477. f ( x ) = ( x ) 3 478. f ( x ) = x 1/3 +(2 x ) 1/3 479. f ( x ) = x 1/3 x 2/3 480. f ( x ) =2sin( x )+sin(2 x )
500. f ′( x ) = x + x 2 , f (0) =2 501. f ′( x ) =cos x +sec 2 ( x ), f ⎛ ⎝ π 4 ⎞ ⎠ =2+ 2 2 502. f ′( x ) = x 3 −8 x 2 +16 x +1, f (0) =0
481. f ( x ) = sec 2 ( x )+1 482. f ( x ) = sin x cos x
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