3020 Differentials and Linear Approximation AB Calculus

Related Rates : How fast is y changing as x is changing?- Differentials: How much does y change as x changes?

Approximation A. Differentials Goal: Answer Two Questions - How much has y changed? and - What is y ‘s new value?

IF… My waist size is 36 inches IF I increases my radius 1 inch, how much larger would my belt need to be ? 2𝜋 𝑖𝑛𝑐ℎ𝑒𝑠 The earths circumference is 24,367.0070904 miles. IF I increases the earth’s radius 1 inch, how much larger would the circumference be ? 2𝜋 𝑖𝑛𝑐ℎ𝑒𝑠

The Change in Value : The Differential The Differential finds a QUANTITY OF CHANGE ! REM: ∆𝑥=4 3 NOTE: Finds the change in y -- NOT the value of y

The Change in Value : The Differential The Differential finds a QUANTITY OF CHANGE ! REM: ∆𝑥=3 9 4 NOTE: Finds the change in y -- NOT the value of y

The Change in Value : The Differential The Differential finds a QUANTITY OF CHANGE ! REM: ∆𝑥=2 6 4 NOTE: Finds the change in y -- NOT the value of y

The Change in Value : The Differential The Differential finds a QUANTITY OF CHANGE ! REM: ∆𝑥= 3 2 9 8 NOTE: Finds the change in y -- NOT the value of y

Differentials and Linear Approximation in the News

Algebra to Calculus dy: The DIFFERENTIAL: “How much has y changed?” “the first difference in y for a fixed change in x ” Notation: dy: Also written: df

The Differential finds a QUANTITY OF CHANGE ! In Calculus dy approximates the change in y using the TANGENT LINE. 1 3 4 3 3 4 3 NOTE: APPROXIMATES the change in y The smaller the ∆𝑥 the better the approximation

The Differential Function: Example 1 Find the Differential Function and use it to approximate change. A). Find the differential function. B). Approximate the change in y at with 𝑑𝑦= cos 𝑥 𝑑𝑥 𝑑𝑦 𝑑𝑥 = cos 𝑥 𝑑𝑦=cos⁡( 𝜋 6 ) 𝜋 36 𝑑𝑦= 3 2 ∗ 𝜋 36 = 𝜋 3 72 ≈.0756

The Differential Function: Example 2 Find the Differential Function and use it to approximate the volume of latex in a spherical balloon with inside radius and thickness A). Find the differential function. B). Approximate the change in V. C). Find the actual Volume. 𝑑𝑉 𝑑𝑟 =4𝜋 𝑟 2 𝑑𝑉=4𝜋 𝑟 2 (𝑑𝑟) 𝑑𝑉=4𝜋 4 2 1 16 =4𝜋≈12.566 4 3 𝜋 𝑟 3 = 4 3 𝜋 65 16 3 − 4 3 𝜋 4 3 280.8463−268.0826=12.764

B: Linearization “Make It Linear!”

L(x) = f(a) + f / (a) (x – a) Linearization: Linearization: y – y1 = m (x – x1 ) y = y1 + m (x – x1 ) L(x) = f(a) + f / (a) (x – a) The standard linear approximation of f at a The point x = a is the center of the approximation

Linearization 𝑦= 3 2 𝑦=sin⁡(𝑥) 𝑦 ′ = 1 2 𝑦 ′ =cos⁡(𝑥) L 𝑥 =𝑦+𝑦′(𝑥−𝑎) 𝑦= 3 2 𝑦=sin⁡(𝑥) 𝑦 ′ = 1 2 𝑦 ′ =cos⁡(𝑥) L 𝑥 =𝑦+𝑦′(𝑥−𝑎) L 𝑥 = 3 2 + 1 2 𝑥− 𝜋 3 𝑥−

Linearization 𝑦= 2𝑥−1 𝑦 5 =3 𝑦 ′ = 1 2 2𝑥−1 −1 2 ∗2 𝑦 ′ = 1 2𝑥−1 𝑦= 2𝑥−1 𝑦 5 =3 𝑦 ′ = 1 2 2𝑥−1 −1 2 ∗2 𝑦 ′ = 1 2𝑥−1 𝑦 ′ 5 = 1 3 𝐿 𝑥 =3+ 1 3 𝑥−5

C: Tangent Line Approximation #1 find friendly # #2 ∆𝑥 What is the new value? #3 y 𝜋 6 = 6𝜋 36 given friendly #4 y’ y2 – y1 = m ( x2 – x1 ) y2 = y1 + m (Δx) 𝜋 4 = 9𝜋 36 #5 values 𝜋 3 = 12𝜋 36

Linear Approximation - Tangent Line Approximation EXAMPLE: . Find friendly # Wants the VALUE! 𝑎= 12𝜋 36 𝑜𝑟 𝜋 3 𝑓 12𝜋 36 + −𝜋 36 ≈ cos 𝑥 +(−sin⁡(𝑥)( −𝜋 36 ) ∆𝑥 𝑔𝑖𝑣𝑒𝑛 −𝑓𝑟𝑖𝑒𝑛𝑑𝑙𝑦 𝑦 𝜋 3 = 1 2 ≈ 1 2 + − 3 2 −𝜋 36 𝑦=cos⁡(𝑥) ≈ 1 2 + 3 𝜋 72 𝑦 ′ =−sin⁡(𝑥) 𝑦 ′ 𝜋 3 = − 3 2 𝑐𝑜𝑠 11𝜋 36 ≈ 1 2 + 3 𝜋 72 ∆𝑥=− 𝜋 36

Linear Approximation - Tangent Line Approximation EXAMPLE: Approximate . Wants the VALUE!

II. Error

ERROR: There are TWO types of error: A. Error in measurement tools - quantity of error - relative error - percent error B. Error in approximation formulas - over or under approximation - Error Bound - formula

A. Error in Measurement Tools

Choose either 1 or 2 0 1 2 .00,000,1 .00,000,2

EXAMPLE 1: Measurement (A) Volume and Surface Area: The measurement of the edge of a cube is found to be 12 inches, with a possible error of 0.03 inches. Use differentials to approximate the maximum possible error in computing:   the volume of a cube the surface area of a cube find the range of possible measurements in parts (a) and (b).  

EXAMPLE 2: Measurement (A) Volume and Surface Area: the radius of a sphere is claimed to be 6 inches, with a possible error of .02 inch Use differentials to approximate the maximum possible error in calculating the volume of the sphere. Use differentials to approximate the maximum possible error in calculating the surface area. Determine the relative error and percent error in each of the above.

EXAMPLE 3: Measurement (B) : Tolerance Area: The measurement of a side of a square is found to be 15 centimeters. Estimate the maximum allowable percentage error in measuring the side if the error in computing the area cannot exceed 2.5%.

EXAMPLE 4: Measurement (B) : Tolerance Circumference The measurement of the circumference of a circle is found to be 56 centimeters. Estimate the maximum allowable percentage error in measuring the circumference if the error in computing the area cannot exceed 3%.  

B. Error in Approximation Formulas

ERROR: Approximation Formulas Error = (actual value – approximation)  either Pos. or Neg. Error Bound = | actual – approximation | For Linear Approximation: The Error Bound formula is Since the approximation uses the TANGENT LINE the over or under approximation is determined by the CONCAVITY (2ndDerivative Test)

In Calculus dy approximates the change in y using the TANGENT LINE. The ERROR depends on distance from center( ) and the bend in the curve ( f ” (x))

Example 5: Approximation Error = (actual value – approximation)  either Pos. or Neg. Error Bound = | actual – approximation | For Linear Approximation: The Error Bound formula is EX: Find the Error in the linear approximation of

Example 6: Approximation EX: Find the Error in the linear approximation of

Last Update: 11/04/11

New Value : Tangent line Approximation In words: _____________________________________________ With the differential :

Algebra to Calculus How much has y changed?

Algebra to Calculus How much has y changed?