Section 4.1 Linear Approximations and Applications MAT 1234 Calculus I Section 4.1 Linear Approximations and Applications http://myhome.spu.edu/lauw

Next WebAssign 4.1

Preview The need for approximations: Formulas can be simplified. Very popular method used in physical sciences.

Preview Introduce a simple approximation method (linear approximation) by using the first derivative of the function. It is a fundamental idea of how computing technology works. Formula  Idea+Evidence  Applications Introduce the concept of differentials

Linear Approximations When 𝑥 is near a point 𝑎, we can approximate the value of 𝑓(𝑥) by Why?

Linear Approximations When 𝑥 is near a point 𝑎, we can approximate the value of 𝑓(𝑥) by ? Easy to find

Linear Approximations When 𝑥 is near a point 𝑎, we can approximate the value of 𝑓(𝑥) by ? Easy to find

Linear Approximations When 𝑥 is near a point 𝑎, we can approximate the value of 𝑓(𝑥) by ? Easy to find

Linear Approximations When 𝑥 is near a point 𝑎, we can approximate the value of 𝑓(𝑥) by ? Easy to find

Linear Approximations When 𝑥 is near a point 𝑎, we can approximate the value of 𝑓(𝑥) by ? Easy to find

Linear Approximations When 𝑥 is near a point 𝑎, we can approximate the value of 𝑓(𝑥) by Why?

𝑦 𝑎 𝑥

𝑦 𝑓(𝑥) 𝑓(𝑎) 𝑎 𝑥

𝑦 𝑎 𝑥 𝑓(𝑥) 𝑓(𝑎)

Example 1 Estimate the value of 9.036 is near 9 Let us consider the function when x is near 9

Step 1: Define the function and the near by point Estimate the value of 9.036 is near 9

Step 2: Find

Step 3: Find the linear approximation

Step 4: Substitute x=9.036 into the approximation in Step 3

Compare this with your calculator! Example 1 Estimate the value of Compare this with your calculator!

Example 1 Remarks Pay attention to the usage of the approximate and equal signs. Correct Incorrect

Example 1 Remarks Pay attention to the usage of the approximate and equal signs.

Expectations You are expected show all 4 steps. Historically, a few students each year skip step 3. In the quiz and exam, I will specially ask “Find the linear approximation of 𝑓 𝑥 .” A few students also ignore 𝑓 9.036 in

Example 2 Estimate the value of

Step 1: Define the function and the near by point Estimate the value of

Step 2: Find

Step 3: Find the linear approximation

Step 4: Substitute x=2.001 into the approximation in Step 3

Compare this with your calculator! Example 2 Estimate the value of Compare this with your calculator!

Better Approximations Taylor Polynomials

Differentials

Differentials 𝑦 𝑎 𝑥 𝑓(𝑥) 𝑓(𝑎)

Differentials 𝑦 𝑎 𝑎+𝑑𝑥

Differentials Suppose 𝑦=𝑓(𝑥) Let 𝑑𝑥 be an independent variable We define a new dependent variable 𝑑𝑦 as There are 2 dependent variables and 2 independent variables

Differentials Suppose 𝑦=𝑓(𝑥) Let 𝑑𝑥 be an independent variable We define a new dependent variable 𝑑𝑦 as There are 2 dependent variables and 2 independent variables

Differentials 𝑦 depends on 𝑥 dy depends on x and dx dx and dy are called differentials f’(x)=dy/dx (This explains the notation ) Use differentials to find anti-derivatives

Differentials 𝑦 depends on 𝑥 𝑑𝑦 depends on 𝑥 and 𝑑𝑥 dx and dy are called differentials f’(x)=dy/dx (This explains the notation ) Use differentials to find anti-derivatives

Differentials 𝑦 depends on 𝑥 𝑑𝑦 depends on 𝑥 and 𝑑𝑥 𝑑𝑥 and 𝑑𝑦 are called differentials f’(x)=dy/dx (This explains the notation ) Use differentials to find anti-derivatives

Differentials 𝑦 depends on 𝑥 𝑑𝑦 depends on 𝑥 and 𝑑𝑥 𝑑𝑥 and 𝑑𝑦 are called differentials 𝑓’(𝑥)=𝑑𝑦/𝑑𝑥 (This explains the notation 𝑑𝑦 𝑑𝑥 )

Differentials 𝑦 depends on 𝑥 𝑑𝑦 depends on 𝑥 and 𝑑𝑥 𝑑𝑥 and 𝑑𝑦 are called differentials 𝑓’(𝑥)=𝑑𝑦/𝑑𝑥 (This explains the notation 𝑑𝑦 𝑑𝑥 ) Use differentials to find anti-derivatives

Example 3

Example 4