Tangent Line Approximations A tangent line approximation is a process that involves using the tangent line to approximate a value. Newton used this method.

Slides:

Advertisements

Similar presentations

Linear Approximation and Differentials

Advertisements

Section Differentials Tangent Line Approximations A tangent line approximation is a process that involves using the tangent line to approximate a.

4.5: Linear Approximations and Differentials

3 Copyright © Cengage Learning. All rights reserved. Applications of Differentiation.

Tangent Planes and Linear Approximations

Sec. 4.5: Integration by Substitution. T HEOREM 4.12 Antidifferentiation of a Composite Function Let g be a function whose range is an interval I, and.

The Derivative. Objectives Students will be able to Use the “Newton’s Quotient and limits” process to calculate the derivative of a function. Determine.

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

DO NOW Find the equation of the tangent line of y = 3sin2x at x = ∏

Derivative of an Inverse AB Free Response 3.

Secant Method Another Recursive Method. Secant Method The secant method is a recursive method used to find the solution to an equation like Newton’s Method.

Who wants to be a Millionaire? Hosted by Kenny, Katie, Josh and Mike.

Mean Value Theorem for Derivatives.

AP CALCULUS AB Chapter 4: Applications of Derivatives Section 4.5:

Lesson 3-11 Linear Approximations and Differentials.

Aim: Differentials Course: Calculus Do Now: Aim: Differential? Isn’t that part of a car’s drive transmission? Find the equation of the tangent line for.

A technique for approximating the real zeros of a Function.

Division and Factors When we divide one polynomial by another, we obtain a quotient and a remainder. If the remainder is 0, then the divisor is a factor.

Determine whether Rolle’s Theorem can be applied to f on the closed interval [a,b]. If Rolle’s theorem can be applied, find all values of c in the open.

Equations Reducible to Quadratic

3.1 Definition of the Derivative & Graphing the Derivative

Differentials Intro The device on the first slide is called a micrometer….it is used for making precision measurements of the size of various objects…..a.

Problem of the Day No calculator! What is the instantaneous rate of change at x = 2 of f(x) = x2 - 2 ? x - 1 A) -2 C) 1/2 E) 6 B) 1/6 D) 2.

4.1 ANTIDERIVATIVES & INDEFINITE INTEGRATION. Definition of Antiderivative  A function is an antiderivative of f on an interval I if F’(x) = f(x) for.

Linearization and Newton’s Method. I. Linearization A.) Def. – If f is differentiable at x = a, then the approximating function is the LINEARIZATION of.

Linear approximation and differentials (Section 2.9)

Suppose we are given a differential equation and initial condition: Then we can approximate the solution to the differential equation by its linearization.

For any function f (x), the tangent is a close approximation of the function for some small distance from the tangent point. We call the equation of the.

Solving Linear Equations Substitution. Find the common solution for the system y = 3x + 1 y = x + 5 There are 4 steps to this process Step 1:Substitute.

4.1 Extreme Values of Functions

Linear Approximation and Differentials Lesson 4.8.

3 Copyright © Cengage Learning. All rights reserved. Applications of Differentiation.

3.9 Differentials. Objectives Understand the concept of a tangent line approximation. Compare the value of a differential, dy, with the actual change.

Miss Battaglia AB/BC Calculus.  /tangent_line/ /tangent_line/ y.

Linearization, Newton’s Method

Differentials A quick Review on Linear Approximations and Differentials of a function of one Variable.

Solving First-Order Differential Equations A first-order Diff. Eq. In x and y is separable if it can be written so that all the y-terms are on one side.

Miss Battaglia BC Calculus. Let y=f(x) represent a functions that is differentiable on an open interval containing x. The differential of x (denoted by.

3.9 Differentials Let y = f(x) represent a function that is differentiable in an open interval containing x. The differential of x (denoted by dx) is any.

3 Copyright © Cengage Learning. All rights reserved. Applications of Differentiation.

Tangent Line Approximations Section 3.9 Notes. Given a function, f (x), we can find its tangent at x = a. The equation of the tangent line, which we’ll.

Section 9.4 – Solving Differential Equations Symbolically Separation of Variables.

Solving Systems of Equations

§ 4.2 The Exponential Function e x.

Linear approximation and differentials (Section 3.9)

Warm-Up- Test in the box…

z = z0 + fx(x0,y0)(x – x0) + fy(x0,y0)(y – y0)

3.1 – Derivative of a Function

3.1 – Derivative of a Function

Techniques of Integration

Problem of the Day (Calculator Allowed)

Copyright © Cengage Learning. All rights reserved.

Katherine Han Period Linear Approximation Katherine Han Period

Linear Approximation and Differentials

Question Find the derivative of Sol..

Section Euler’s Method

Secant Method – Derivation

8. Linearization and differentials

Copyright © Cengage Learning. All rights reserved.

Copyright © Cengage Learning. All rights reserved.

Copyright © Cengage Learning. All rights reserved.

Linearization and Newton’s Method

4.5: Linear Approximations, Differentials and Newton’s Method

Linear approximation and differentials (Section 3.9)

Copyright © Cengage Learning. All rights reserved.

MATH 1910 Chapter 3 Section 8 Newton’s Method.

3.10 Linear Approximations and Differentials

Definition of logarithm

Presentation transcript:

Tangent Line Approximations A tangent line approximation is a process that involves using the tangent line to approximate a value. Newton used this method to approximate zeros of functions. Another use of this method is to approximate the value of a function.

Approximating the Value of a Function To approximate the value of f at x = c, f(c): 1.Choose a convenient value of x close to c, call this value x 1. 2.Write the equation of the tangent line to the function at x 1. y – f(x 1 ) = f ’(x 1 )( x - x 1 ) 3.Substitute the value of c in for x and solve for y. 4.f(c) is approximately equal to this value of y.

Example 1 If, use a tangent line approximation to find f(1.2). 1.Since c = 1.2, choose x 1 = 1, which is close to Since f(1) = 4 and f’(x) = 2x gives f’(1) = 2, then eq. of tangent line is y – 4 = 2(x – 1) 3.Substitute 1.2 in for x and y = 4 + 2(.2) = Therefore, since 1 is close to 1.2, f(1.2) is close to Check this with the actual value f(1.2) = 4.44

Example 2 Approximate sqrt(18) using the tangent line approximation method (also called using differentials). 1.Choose x 1 = 16 and let f(x) = sqrt(x), then f (x 1 ) = 4 and f’(x 1 ) = 1/(2 * sqrt(16)) = 1/8 2.Eq. of Tangent: y – 4 = (1/8)(x – 16) 3.When x = 18, y = 4 + (1/8)(2) = 4 + ¼ = Therefore sqrt(18) is approximately Check with sqrt(18) on your calculator: 4.243…

Definition of Differentials Let y = f(x) represent a function that is differentiable in an open interval containing x. Since then Therefore dx is called the differential of x and is equal to  x. dy is called the differential of y and is approximately equal to  y as long as  x is close to zero.

Example 3 Find both  y and dy and compare. y = 1 – 2x 2 at x = 1 when  x = dx = -.1 Solution:  y = f(x +  x) – f(x) = f(.9) – f(1) = -.62 – (-1) =.38 dy = f’(x)dx = (-4)(-.1) =.4 ** Note that  y ≈ dy

Example 4 The radius of a ball bearing is measured to be.7 inches. If the measurement is correct to within.01 inch, estimate the propagated error in the volume V of the ball bearing. **Propagated error means the resulting change, or error, in measurement

Example 4 Continued To decide whether the propagated error is small or large, it is best looked at relative to the measurement being calculated. Find the relative error in volume of the ball bearing. ** Relative error is dy/y, or in this case dV/V Find the percent error, (dV/V) * 100.