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Tangent lines

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)).

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Tangent lines

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  1. 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)).

  2. Tangent lines • Ex. 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 is or simplifies to

  3. Velocities • Recall: instantaneous velocity is limit of average velocity • Suppose 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

  4. Rates of change • Let 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

  5. Definition of derivative • Definition 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 and they are the same.

  6. Example • Ex. Find the derivative given Sol. Since does not exist, the derivative does not exist.

  7. Example • Ex. Determine the existence of of f(x)=|x|. Sol. Since does not exist.

  8. Continuity and derivative • Theorem If exists, then f(x) is continuous at x0. Proof. • Remark. The continuity does not imply the existence of derivative. For example,

  9. Interpretation of derivative • The 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.

  10. Derivative as a function • Recall 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 derivative exists, we can regard as a function which assigns to x.

  11. Remark • Some other limit forms

  12. Example Find the derivative function of Sol. Let a be any number, by definition, Letting a vary, we get the derivative function

  13. Other notations for derivative • If 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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