Download Math 129 Homework - Linearization and Newton`s Method

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);