Estimation newtons method and areas try to compute ln2

Question

Estimation: Newton's method and areas - Math 1A, section 103

1. Approximate √3 using Newton's method.

2. Try to compute ln(2) without a calculator. In other words, use Newton's method to approximate the solutions of the equation e^x - 2 = 0.

3. Approximate π.

4.  (a) Find an expression for the area under the curve y = x³ from 0 to 1 as a limit.

(b) Evaluate the limit in part (a). You may use the formula

1³ + 2³ + 3² + · · · + n³ = (n(n + 1)/2)².

5. Using the definition of the definite integral in terms of limits of Riemann sums, prove that

_a∫^b x dx = (b² - a²)/2.

5. Use the fundamental theorem of calculus to find

_-1∫² x³ - 2x dx.