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Math 129 Homework - Linearization and Newton’s Method 1. a) Find the linear approximation (use local linearization) to f ( x) 3 x near x=8. b) Use the approximation in (a) to estimate 3 7 . 2. Use Newton’s method to solve x3 = 7. (Use xo = 2.) Note that the solution obtained should be an estimate of 3 7 . Compare your result to 1b. 3. Use Newton’s method to solve the equation (find all solutions): a) x + ln(x) = 2 b) x3 3x 9 c) x3 3x 2 1 4. Consider f ( x) x3 3x 2 x 1 . a) On your calculator, graph the function and observe that there is a zero on the interval [2, 3] b) Start xo = 1 and use Newton’s method to approximate the zero of the function. c) Start xo = 2 and use Newton’s method to approximate the zero of the function. d) Compare the results of (b) and (c). Explain clearly what’s going on! 1 near x = 2. 2 x 8 4 and . 4.08 3.96 5. Find the tangent line approximation to Use your result to estimate 6. Find the linear approximation to f ( x) 2 x 9 near x=0; 7. Use a linear approximation to estimate: a) 4 16.08 ; b) ln(1.01) and, ln(1.07);