Calculus Volume 1

484

Chapter 4 | Applications of Derivatives

For the following exercises, use Newton’s method to find the location of the local minima and/or maxima of the following functions; round to three decimals.

454. Why would you use the secant method over Newton’s method? What are the necessary restrictions on f ? For the following exercises, use both Newton’s method and the secant method to calculate a root for the following equations. Use a calculator or computer to calculate how many iterations of each are needed to reach within three decimal places of the exact answer. For the secant method, use the first guess from Newton’s method. 455. f ( x ) = x 2 +2 x +1, x 0 =1 456. f ( x ) = x 2 , x 0 =1 457. f ( x ) = sin x , x 0 =1 458. f ( x ) = e x −1, x 0 =2 459. f ( x ) = x 3 +2 x +4, x 0 =0 In the following exercises, consider Kepler’s equation regarding planetary orbits, M = E − ε sin( E ), where M is the mean anomaly, E is eccentric anomaly, and ε measures eccentricity. 460. Use Newton’s method to solve for the eccentric anomaly E when the mean anomaly M = π 3 and the eccentricity of the orbit ε =0.25; round to three decimals. 461. Use Newton’s method to solve for the eccentric anomaly E when the mean anomaly M = 3 π 2 and the eccentricity of the orbit ε =0.8; round to three decimals. The following two exercises consider a bank investment. The initial investment is $10,000. After 25 years, the investment has tripled to $30,000. 462. Use Newton’s method to determine the interest rate if the interest was compounded annually. 463. Use Newton’s method to determine the interest rate if the interest was compounded continuously. 464. The cost for printing a book can be given by the equation C ( x ) =1000+12 x + ⎛ ⎝ 1 2 ⎞ ⎠ x 2/3 . Use Newton’s method to find the break-even point if the printer sells each book for $20.

438. Minimum of f ( x ) = x 2 +2 x +4 439. Minimum of f ( x ) =3 x 3 +2 x 2 −16 440. Minimum of f ( x ) = x 2 e x

441. Maximum of f ( x ) = x + 1 x 442. Maximum of f ( x ) = x 3 +10 x 2 +15 x −2 443. Maximum of f ( x ) = x − x 3 x 444. Minimum of f ( x ) = x 2 sin x , closest non-zero minimum to x =0 445. Minimum of f ( x ) = x 4 + x 3 +3 x 2 +12 x +6 For the following exercises, use the specified method to solve the equation. If it does not work, explain why it does not work. 446. Newton’s method, x 2 +2=0 447. Newton’s method, 0= e x 448. Newton’s method, 0=1+ x 2 starting at x 0 =0 449. Solving x n +1 =− x n 3 starting at x 0 =−1 For the following exercises, use the secant method, an alternative iterative method to Newton’s method. The formula is given by x n = x n −1 − f ( x n −1 ) x n −1 − x n −2 f ( x n −1 )− f ( x n −2 ) . 450. Find a root to 0= x 2 − x −3 accurate to three decimal places. 451. Find a root to 0= sin x +3 x accurate to four decimal places. 452. Find a root to 0= e x −2 accurate to four decimal places. 453. Find a root to ln( x +2) = 1 2 accurate to four decimal places.

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