Learning Outcomes
- Find the linear approximation to a function at a point.
- Calculate the differential of a given function.
Textbook
- Chapter 4.2 Linear Approximation and Differentials
Textbook Assignment
- p. 364: 62, 63, 67-70 all, 72-74 all
WeBWork Assignment
- Application-Linearization
- Application-Differentials
Exit problems of the session
-
Find the linear approximation to $y=\tan x$ near $a=\dfrac{\pi}{4}$.
-
Use appropriate linear approximation to estimate $\sqrt[3]{26.95}$.
- Find the differential $dy$ for $y=xe^{-x}$ and evaluate it at $x=1$ and $dx=0.02$.
Key Concepts
- A differentiable function $y=f(x)$ can be approximated at $a$ by the linear function:
$L(x)=f(a)+f'(a)(x-a)$
- The differential $dx$ is an independent variable that can be assigned any nonzero real number; the differential $dy$ is define to be
$dy=f'(x)dx$
- For a function $y=f(x)$, if $x$ changes from $a$ to $a+dx$, then $dy=f'(x)dx$ is an approximation for the change in $y$. The actual change in $y$ is $\Delta y=f(a+dx)-f(a)$.
Videos and Practice Problems of Selected Topics
- $\rhd$ Local linearity (9:37) Estimate $\sqrt{4.36}$.
- $\rhd$ Linear approximation of a rational function (7:10) Find a linear expression that approximates $\dfrac{y}{x-1}$ around $x=-1$.
- $\rhd$ Comparing $\Delta y$ and $dy$ (4:07) Let $y=2x^2$.
- Find the change in $y$, $\Delta y$, when $x=2$ and $\Delta x=0.3$.
- Find differential $dy$ when $x=2$ and $dx=0.3$.
