Tangent Line Approximation and Implicit Differentiation, Study notes of Calculus

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3.10 (#35)

Tangent line approximation

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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

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

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Estimate the cubic root of 8.

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

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

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differentials

dy

=f’(x)

dx

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

7. (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)