Question Find the derivative of Sol..

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Presentation transcript:

Question Find the derivative of Sol.

Question Find

Question For what values of the function (1) is continuous at 0, (2) is differentiable at 0, (3) is continuous at 0. Sol. (1) (2) (3)

Hyperbolic functions Definition of the hyperbolic functions

Hyperbolic functions Derivatives of hyperbolic functions

Differentials The differential of the independent variable x, denoted by dx, is the increment If y=f(x) is a differentiable function, the differential of the function, denoted by dy, is Generally, the differential dy is not equal to the increment

Linear approximations An equation of the tangent line to y=f(x) at the point (a,f(a)) is If the graph of f is smooth, we can approximate the curve using the tangent line in the neighborhood of a, i.e., This is called the linear approximation of f at a.

Example Find the linear approximation of at a=1 and use it to approximate the numbers and Are these approximations overestimates or underestimates? Sol. They are overestimates because the tangent line lies above the curve.

Error estimation of linear approximation Suppose f is differentiable at a, then its linear approximation at a, Consider the error It can be easily proved by the definition of derivative that which means that the error is much smaller then

Error estimation of linear approximation Some notations: increment of x at corresponding increment of y at We proved

Example Note that only when x is close to a, the approximation of f(x) by L(x) is accurate enough. Ex. Find an approximate value of Sol. Since we use the linear approximation So

Question Question. Find an approximation of Sol.

Differentials In the linear approximation we call the differential of f at and denote Letting f(x)=x, we see That is, the increment of x is the same as its differential. but generally not the same The formula justifies the notation The linear approximation is using to approximate

Example Ex. Compare the values of and if and x changes from 2 to 2.01. Sol. When |x| is small, we have the following approximation, which can be derived by replacing increment by differential.

Homework 7 Section 3.11: 24, 32, 36, 39 Review exercises: 14, 26, 53, 100, 101, 102, 103, 105