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3020 Differentials and Linear Approximation BC Calculus

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

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

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

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Algebra to Calculus How much has y changed?

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

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Differentials and Linear Approximation in the News

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Algebra to Calculus How much has y changed?

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

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The Differential finds a QUANTITY OF CHANGE ! In Calculus dy approximates the change in y using the TANGENT LINE. NOTE: APPROXIMATES the change in y

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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

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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.

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B: Linearization “Make It Linear!”

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Linearization: y – y 1 = m (x – x 1 ) y = y 1 + m (x – x 1 ) The standard linear approximation of f at a The point x = a is the center of the approximation L(x) = f(a) + f / (a) (x – a)

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Linearization

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C: Tangent Line Approximation What is the new value? y – y 1 = m ( x – x 1 ) y 2 – y 1 = m ( x 2 – x 1 ) y 2 = y 1 + m (Δx)

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

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

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

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II. Error

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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

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A. Error in Measurement Tools

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01 2 Choose either 1 or 2

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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).

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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.

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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%.

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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%.

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B. Error in Approximation Formulas

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

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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))

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

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

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Last Update: 11/04/10

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Section 3.9 - Differentials. Local Linearity If a function is differentiable at a point, it is at least locally linear. Differentiable.

Section 3.9 - Differentials. Local Linearity If a function is differentiable at a point, it is at least locally linear. Differentiable.

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