5. Tangent Planes and Chain Rule

Linear Approximation — Quiz

Test your understanding of linear approximation with 5 practice questions.

Practice Questions

Question 1

What is the standard linear approximation of a differentiable function $f(x,y)$ near the point $(a,b)$?

Question 2

Use linear approximation for $f(x,y)=x^2+y^2$ at $(1,2)$ to estimate $f(1.1,1.9)$.

Question 3

The linear approximation of $f(x,y)=\ln x+\ln y$ at $(1,1)$ is used to estimate $f(1.02,0.98)$. What value is obtained?

Question 4

For $f(x,y,z)=xyz$, use the total differential at $(1,2,3)$ to approximate the new value when $(x,y,z)$ changes by $(0.02,-0.01,0.01)$.

Question 5

Why is a linear approximation usually most accurate when the input point is close to the base point?