Let y=f(x) be a differentiable function such that (dy)/(dx)=(x)/(y) and f(8)=2. What is the approximation of f(8.1) using the line tangent to the graph of f at x=8 ?

Identify derivative & point: Identify the derivative and the point of interest. Given that (dy)/(dx) = (x)/(y) and f(8) = 2, we need to find the derivative at x = 8.Calculate derivative at x=8: Calculate the derivative at x = 8. (dy)/(dx) at x = 8 is (8)/(2) = 4.Use point-slope form: Use the point-slope form of the equation of a line to find the tangent line. The slope of the tangent line at x = 8 is 4, and the point on the curve is (8, 2). Equation of the tangent line: y - 2 = 4(x - 8).Simplify tangent line equation: Simplify the equation of the tangent line. y - 2 = 4x - 32, y = 4x - 30.Approximate f(8.1): Use the tangent line to approximate f(8.1). Substitute x = 8.1 into the tangent line equation: y = 4(8.1) - 30, y = 32.4 - 30, y = 2.4.

Home

Math Problems

Calculus

Linear approximation

Full solution

Q. Let

y=f(x)

be a differentiable function such that

\frac{d y}{d x}=\frac{x}{y}

and

f(8)=2

. What is the approximation of

f(8.1)

using the line tangent to the graph of

f

x=8

Identify derivative & point: Identify the derivative and the point of interest. $\newline$ Given that $\frac{dy}{dx} = \frac{x}{y}$ and $f(8) = 2$ , we need to find the derivative at $x = 8$ .
Calculate derivative at $x=8$ : Calculate the derivative at $x = 8$ . $\frac{dy}{dx}$ at $x = 8$ is $\frac{8}{2} = 4$ .
Use point-slope form: Use the point-slope form of the equation of a line to find the tangent line. $\newline$ The slope of the tangent line at $x = 8$ is $4$ , and the point on the curve is $(8, 2)$ . $\newline$ Equation of the tangent line: $y - 2 = 4(x - 8)$ .
Simplify tangent line equation: Simplify the equation of the tangent line. $\newline$ $y - 2 = 4x - 32$ , $\newline$ $y = 4x - 30$ .
Approximate $f(8.1)$ : Use the tangent line to approximate $f(8.1)$ . $\newline$ Substitute $x = 8.1$ into the tangent line equation: $\newline$ $y = 4(8.1) - 30$ , $\newline$ $y = 32.4 - 30$ , $\newline$ $y = 2.4$ .