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Tangent lines Recall: tangent line is the limit of secant line The tangent line to the curve y=f(x) at the point P(a,f(a)) is the line through P with slope provided that the limit exists. Remark. If the limit does not exist, then the curve does not have a tangent line at P(a,f(a)). h a f h a f a x a f x f m h a x ) ( ) ( lim ) ( ) ( lim 0

# Tangent lines

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Tangent lines. Recall: tangent line is the limit of secant line The tangent line to the curve y=f(x) at the point P(a,f(a)) is the line through P with slope provided that the limit exists. Remark. If the limit does not exist, then the curve does not have a tangent line at P(a,f(a)). - PowerPoint PPT Presentation

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• Tangent linesRecall: tangent line is the limit of secant lineThe tangent line to the curve y=f(x) at the point P(a,f(a)) is the line through P with slope

provided that the limit exists.Remark. If the limit does not exist, then the curve does not have a tangent line at P(a,f(a)).

• Tangent linesEx. Find an equation of the tangent line to the hyperbola y=3/x at the point (3,1). Sol. Since the limit

an equation of the tangent line isor simplifies to

• Velocities Recall: instantaneous velocity is limit of average velocitySuppose the displacement of a motion is given by the function f(t), then the instantaneous velocity of the motion at time t=a is

Ex. The displacement of free fall motion is given by find the velocity at t=5.Sol. The velocity is

• Rates of changeLet The difference quotient

is called the average rate of change of y with respect to x. Instantaneous rate of change =

Ex. The dependence of temperature T with time t is given by the function T(t)=t3-t+1. What is the rate of change of temperature with respective to time at t=2?Sol. The rate of change is

• Definition of derivativeDefinition The derivative of a function f at a number a,denoted by is

if the limit exists.

Similarly, we can define left-hand derivative and right-hand derivative exists if and only if both and exist andthey are the same.

• ExampleEx. Find the derivative given

Sol. Since does not exist,

the derivative does not exist.

• ExampleEx. Determine the existence of of f(x)=|x|. Sol. Since

does not exist.

• Continuity and derivativeTheorem If exists, then f(x) is continuous at x0.

Proof.

Remark. The continuity does not imply the existence of derivative.

For example,

• Interpretation of derivativeThe slope of the tangent line to y=f(x) at P(a,f(a)), is the derivative of f(x) at a,

The derivative is the rate of change of y=f(x) with respect to x at x=a. It measures how fast y is changing with x at a.

• Derivative as a functionRecall that the derivative of a function f at a number a is given by the limit:

Let the above number a vary in the domain. Replacing a by variable x, the above definition becomes

If for any number x in the domain of f, the derivativeexists, we can regard as a function which assigns to x.

• RemarkSome other limit forms

• ExampleFind the derivative function of

Sol. Let a be any number, by definition,

Letting a vary, we get the derivative function

• Other notations for derivativeIf we use y=f(x) for the function f, then the following notations can be used for the derivative:

D and d/dx are called differentiation operators.

A function f is called differentiable at a if exists. f is differentiable on [a,b] means f is differentiable in (a,b) and both and exist.

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