Local linearity
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Local Linearity. The idea is that as you zoom in on a graph at a specific point, the graph should eventually look linear. This linear portion approximates the tangent line to the curve at the specific point.

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Local linearity
Local Linearity

  • The idea is that as you zoom in on a graph at a specific point, the graph should eventually look linear.

  • This linear portion approximates the tangent line to the curve at the specific point.

  • Most of the functions you will encounter are "nearly linear" over very small intervals; that is most functions are locally linear.

  • All functions that are differentiable at some point where x=c are well-modeled by a unique tangent line in a neighborhood of cand are thus considered locally linear.


A function is differentiable if it has a derivative everywhere in its domain. It must be continuous and smooth. Functions on closed intervals must have one-sided derivatives defined at the end points.

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To be differentiable, a function must be continuous and smooth.

Derivatives will fail to exist at:

corner

cusp

discontinuity

vertical tangent


Differentiability vs continuity
Differentiability vs Continuity

Theorem:

  • If f is differentiable at a, then f is continuous at a.

  • *The converse to this theorem is false.

    • A continuous function is not necessarily differentiable.


The derivative is the slope of the original function.

The derivative is defined at the end points of a function on a closed interval.


Warning:

The calculator may return an incorrect value if you evaluate a derivative at a point where the function is not differentiable.

Examples:

nDeriv(1/x,x,0) returns 1000000 on a TI-84, but -∞on a TI-89

nDeriv(abs(x),x,0) returns 0 on a TI-84, but +/-1 on a TI-89


Graphers calculate numerical derivatives by using

difference quotients (the same way you’ve been doing it).

Forward difference quotient:

Backward difference quotient:

Symmetric difference quotient:


If a and b are any two points in an interval on which f is differentiable, then takes on every value between and .

Between a and b, must take on every value between and .

Intermediate Value Theorem for Derivatives

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Given f x 2x 3 3x 2 6x 2
Given f(x) = 2x3–3x2 – 6x+2

  • Plot the derivative

  • What type of function is the derivative?

    • Quadratic (one degree less than f(x))

  • When the derivative is 0, what is happening to the graph of the function?

    • There is either a local max or local min

    • At x = -1 (local max: f(x) changes from increasing to decreasing)

    • At x = 2 (local min : f(x) changes from decreasing to increasing)


Let g x f x 5
Let g(x) = f(x) + 5

  • Graph in Y3

  • Graph the derivative of g in Y4

  • How does the derivative of f compare to the derivative of g?

    • The derivative is the same. The vertical shift does not alter the rate of change. Thus, the derivative remains the same.


Increasing decreasing test
Increasing/Decreasing Test

  • If f '(x) is > 0 on an interval, then f is increasing on that interval.

  • If f '(x) is < 0 on an interval, then f is decreasing on that interval.


Suppose that c is a critical number of a continuous function f
Suppose that c is a critical number of a continuous function f:

  • If f ' changes from positive to negative at c, then f has a local maximum at c.

  • If f ' changes from negative to positive at c, then f has a local minimum at c.

  • (x = c is a critical number of the function f(x) if f ' (c) = 0 or f ' (c) does not exist.


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