Question

If y=5u2+3u−1 and u=x2+5 /18, find dx/dy when x=2.

Find the interval of convergence for the power series ∑n=0[infinity](−1)n64n nx2n Give your answer using interval notation. If you need to use [infinity], type INF. If there is only one point in the interval of convergence, the interval notation is [a]. For example, if 0 is the only point in the interval of convergence, you would answer with [0]. Use Eq. (1) from the text to expand the function into a power series with center c=0 and determine the set of x for which the expansion is valid. f(x)=5+x41 5+x41=∑n=0[infinity] The interval of convergence is:

Answer 1

To find dx/dy when x=2, differentiate u=x^2+5/18. Then substitute the values into the equation. To find the interval of convergence for the power series, use the ratio test.

Step-by-step explanation:

To find dx/dy when x=2 in the equation y=5u^2+3u-1 with u=(x^2+5)/18, we need to find du/dx first, then substitute the values into the equation. Let's find du/dx first:

Given that u=(x^2+5)/18, we can differentiate both sides with respect to x to find du/dx:

du/dx = (2x)/18

Substituting the values into the equation:

y = 5[(2(2)^2)/18]^2 + 3[(2(2)^2)/18] - 1

Simplifying the equation will give you the value of dx/dy when x = 2.

To find the interval of convergence for the power series ∑n=0∞ ((-1)^n*64^n*nx^(2n)), we can use the ratio test. The ratio test states that a power series ∑an(x-a)^n converges if the limit as n approaches infinity of |(an+1(x-a)^(n+1))/an(x-a)^n| is less than 1. Applying the ratio test to our power series, we get:

|((-1)^(n+1)*64^(n+1)*(n+1)*x^(2(n+1)))/(64^n*n*x^(2n))| < 1

Simplifying the equation will give you the interval of convergence.