Complete two iterations of Newton's Method for the function f x( ) = x. 2 - 7 using the initial guess 2.6. Round your answers to four decimal places. ...
3.8, 3.9 Calculus Review Sheet Short Answer 1. Complete two iterations of Newton's Method for the function f (x ) x 2 7 using the initial guess 2.6. Round your answers to four decimal places. 2. Complete two iterations of Newton's Method for the function f (x ) cos x using the initial guess x 1 1.3. Round all numerical values in your answer to four decimal places. 3. Use Newton's Method to approximate the zero(s) of the function f (x ) x 5 4x 1 accurate to three decimal places. 4. Use Newton's Method to approximate the zero(s) of the function f (x ) x 2 x 2 accurate to three decimal places. 5. Approximate the positive zero(s) of the function f (x ) x 3 cos x to three decimal places. Use Newton's Method and continue the process until two successive approximations differ by less than 0.001. 6. Use Newton´s Method to approximate the x-value of the indicated point of intersection of the two graphs accurate to three decimal places.Continue the process until two successive approximations differ by less than 0.001. [Hint: Let h (x ) f (x ) g (x ) .]
f (x ) 3x 1 g (x )
x5
1
Name: ________________________
ID: A
7. A manufacturer of digital audio players estimates that the profit for selling a particular model is P 76x 3 4830x 3 320,000,00 x 60, where P is the profit in dollars and x is the advertising expense in 10,000's of dollars (see figure). Find the smaller of two advertising amounts that yield a profit P of $2,100,000. Round your answer to the nearest dollar.
8. Find the equation of the tangent line T to the graph of f (x )
19 at the given point x2
ÁÊÁ 19 ˆ˜˜ ÁÁ 2, ˜˜ . Á 4 ˜ ¯ Ë
9. Find the equation of the tangent line T to the graph of f (x ) 8 sinx at the given point ÊÁË 6,8sin6 ˆ˜¯ . 10. Compare dy and y for y 4x 2 3 at x = –1 with x dx 0.06. Give your answers to four decimal places. 11. Find the differential dy of the function y x 2 3x 2.
1 inch. Use 78 differentials to approximate the possible propagated error in computing the area of the square.
12. The measurement of the side of a square floor tile is 14 inches, with a possible error of
13. The measurements of the base and altitude of a triangle are found to be 54 and 33 centimeters. The possible error in each measurement is 0.25 centimeter. Use differentials to estimate the propagated error in computing the area of the triangle. Round your answer to four decimal places.
2
ID: A
3.8, 3.9 Calculus Review Sheet Answer Section SHORT ANSWER 1. ANS: 2.6462, 2.6458 PTS: 1 DIF: Medium REF: 3.8.1 OBJ: Estimate a zero of a function using two iterations of Newton's Method MSC: Skill NOT: Section 3.8 2. ANS: f ÁÊË x n ˜ˆ¯ f ÁÊË x n ˜ˆ¯ Ê ˆ Ê ˆ Á ˜ Á ˜ f Ëxn ¯ n xn f Ëxn ¯ xn Ê ˆ f ÁÊË x n ˜ˆ¯ f ÁË x n ˜¯
1
1.3
0.2675
0.9636
0.2776
1.5776
2
1.5776
0.0068
1.0000
0.0068
1.5708
PTS: 1 DIF: Easy REF: 3.8.3 OBJ: Estimate a zero of a function using two iterations of Newton's Method MSC: Skill NOT: Section 3.8 3. ANS: 0.250 PTS: 1 DIF: Medium REF: 3.8.8 OBJ: Estimate a zero of a function using Newton's Method NOT: Section 3.8 4. ANS: 5.464 PTS: 1 DIF: Medium REF: 3.8.10 OBJ: Estimate a zero of a function using Newton's Method NOT: Section 3.8 5. ANS: 0.865 PTS: 1 DIF: Medium REF: 3.8.14 OBJ: Estimate a zero of a function using Newton's Method NOT: Section 3.8 6. ANS: 0.444
MSC: Skill
MSC: Skill
MSC: Skill
PTS: 1 DIF: Medium REF: 3.8.15 OBJ: Estimate the intersection point of two graphs using Newton's Method MSC: Skill NOT: Section 3.8
1
ID: A 7. ANS: $315,369 PTS: 1 DIF: Medium REF: 3.8.40 OBJ: Estimate a zero of a function using Newton's Method in applications MSC: Application NOT: Section 3.8 8. ANS: 19x 57 y 4 4 PTS: 1 DIF: Easy REF: 3.9.2 OBJ: Write an equation of a line tangent to the graph of a function at a specified point MSC: Skill NOT: Section 3.9 9. ANS: y (8cos 6) (x 6 ) 8sin6 PTS: 1 DIF: Easy REF: 3.9.5 OBJ: Write an equation of a line tangent to the graph of a function at a specified point MSC: Skill NOT: Section 3.9 10. ANS: dy 0.4800; y 0.4944 PTS: 1 DIF: Medium REF: 3.9.8 OBJ: Compare the change of y to the differential of y at a given point MSC: Skill NOT: Section 3.9 11. ANS: (2x 3)dx PTS: 1 DIF: Medium REF: 3.9.11 OBJ: Calculate the differential of y for a given function NOT: Section 3.9 12. ANS: 14 square inches 39 PTS: 1 DIF: Easy REF: 3.9.27 OBJ: Estimate the propagated error using differentials NOT: Section 3.9 13. ANS: 10.875 square centimeters PTS: 1 DIF: Medium REF: 3.9.28 OBJ: Estimate the propagated error using differentials NOT: Section 3.9