Exam 11: Taylor Polynomials and Infinite Series

Below is a graph of the functions h(x) and g(x). In using the Newton-Raphson algorithm to find where $h ( x ) = g ( x )$ , which of the following statements is false?

Sum an appropriate infinite series to find the rational number whose decimal expansion is: $0.185 \overline { 185 }$ Enter just a reduced fraction of form $\frac { a } { b }$ .

Which of the following series converge?

Find the second Taylor polynomial of f(x) = sin $x ^ { 2 }$ at x = 0 and use it to approximate the area under the curve f(x) between 0 and $\frac { \pi } { 2 }$ . Enter an unlabeled polynomial in x in standard form followed by a comma and then just a quotient representing the area (π in the numerator).

Use two repetitions of the Newton-Raphson algorithm to approximate the value of x for which $e ^ { x }$ = 3x. Use $x = 0$ as the first approximation. Enter just a real number rounded off to two decimal places.

Determine the sum of the following geometric series: $\frac { 2 ^ { 2 } } { 5 ^ { 3 } }$ + $\frac { 2 ^ { 4 } } { 5 ^ { 5 } }$ + $\frac { 2 ^ { 6 } } { 5 ^ { 7 } }$ + $\frac { 2 ^ { 8 } } { 5 ^ { 9 } }$ + $\frac { 210 } { 5 ^ { 11 } }$ + ... . Enter just a reduced fraction of form $\frac { a } { b }$ .

Use two repetitions of the Newton-Raphson algorithm to approximate $\sqrt { 15 }$ . Enter just a real number rounded off to two decimal places.

Which of the following series converge?

Below is a graph of the function f(x). Which of the following could be the first Taylor polynomial of f(x) at $x = 0 ?$

Let f(x) = ln x. Find the second Taylor polynomial of f(x) at x = 2.

Determine the second Taylor polynomial of sin $x ^ { 2 }$ at x = 0. Enter an unlabeled polynomial in x in standard form (i.e., highest powers first).

Find the Taylor series at x = 0 of f(x) = $e ^ { 3 x }$ by computing four derivatives and using the definition of the Taylor series. Is $1 - 3 x + \frac { 3 ^ { 2 } } { 2 ! } x ^ { 2 } - \frac { 3 ^ { 3 } } { 3 ! } x ^ { 3 } + \frac { 3 ^ { 4 } } { 4 ! } x ^ { 4 } - \ldots$ the correct answer?