Question

Use difference quotients with delta x = 0.1 and delta y = 0.1 to estimate fx(1, 2) and fy(1, 2). (Round your answer to four decimal places.)

f(x,y) = e⁻ˣ siny

Answer 1

To estimate fx(1, 2) and fy(1, 2), we use difference quotients with delta x = 0.1 and delta y = 0.1. For fx(1, 2), the approximate value is 0.0461, and for fy(1, 2), the approximate value is 0.6543.

Step-by-step explanation:

To estimate fx(1, 2) and fy(1, 2) using difference quotients, we need to substitute the values of x=1 and y=2 into the given function f(x,y) = e^(-x)sin(y). Let's start by calculating fx(1,2) using the first-order partial derivative formula:

fx(1,2) = [f(1+0.1, 2) - f(1,2)] / 0.1

Substituting the values and evaluating the function, we get:

fx(1,2) = [e^(-1.1)sin(2) - e^(-1)sin(2)] / 0.1

Calculating the expression, fx(1,2) is approximately equal to 0.0461.

To estimate fy(1,2), we use a similar approach with the second-order partial derivative formula:

fy(1,2) = [f(1, 2+0.1) - f(1,2)] / 0.1

Substituting the values and evaluating the function, we get:

fy(1,2) = [e^(-1)sin(2.1) - e^(-1)sin(2)] / 0.1

Calculating the expression, fy(1,2) is approximately equal to 0.6543.