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Tangent Line Approximation and Implicit Differentiation, Study notes of Calculus

Instructions and examples on the concept of tangent line approximation and its application in finding linearizations of functions and estimating values. It also covers implicit differentiation and its use in solving problems involving curves and particles moving along them.

What you will learn

  • How do you estimate the value of a function using the tangent line approximation?
  • How do you find the linearization of a function at a given point?
  • Given a cone with a fixed total surface area, how do you find the new base radius when changing the height?
  • What is the definition of the tangent line approximation of a function?
  • What is implicit differentiation and how is it used?

Typology: Study notes

2021/2022

Uploaded on 09/27/2022

birkinshaw
birkinshaw 🇺🇸

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1/15 3.10 (#35)2015-11-23 12:21:56
3.10
Tangent line approximation
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1/

3.10 (#35)

Tangent line approximation

2/

Given a function f(x), the function

g(x)= f(a)+f’(a)(x

a)

Is called the linearization , or linear approximation or tangent line approximation of f at a

Discuss upper/lower estimates

4/

Find the linearization of f(x)=sin(x) at 0.

5/

Estimate the cubic root of 8.

7/

Estimate the solution of x^3+2x=0.

8/

(0.05)^3 +2* 0.05 = 0.

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11/

differentials

dy

=f’(x)

dx

2015-11-23 12:21:57 13/15 3.10 (13/15)

  1. (12 points) A particle is traveling along a curve with parametric equations x = x(t), y = y(t). The implicit equation of the curve is y^2 = x^3 + 3x. At time t = 0, the particle is located at the point (1, −2) and its vertical velocity dy dt is 2 units/sec.

Use the tangent line approximation to estimate the location of the particle at time t = 0. 1.

2015-11-23 12:21:57 14/15 3.10 (14/15)