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I’m going nuts over derivatives!!!

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2.1 The Derivative and the Tangent Line Problem. I’m going nuts over derivatives!!!. Calculus grew out of 4 major problems that European mathematicians were working on in the seventeenth century. 1. The tangent line problem. 2. The velocity and acceleration problem.

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Presentation Transcript
slide1
2.1

The Derivative and the

Tangent Line Problem

I’m going nuts

over derivatives!!!

slide2
Calculus grew out of 4 major problems that

European mathematicians were working on

in the seventeenth century.

1. The tangent line problem

2. The velocity and acceleration problem

3. The minimum and maximum problem

4. The area problem

slide3
The tangent line problem

secant line

(c, f(c))

(c, f(c)) is the point of tangency and

f(c+ ) – f(c)

x

is a second point on the graph of f.

slide4
The slope between these two points is

Definition of Tangent Line with Slope m

slide5
Find the slope of the graph of f(x) = x2 +1 at

the point (-1,2). Then, find the equation of the

tangent line.

(-1,2)

slide6
f(x) = x2 + 1

Therefore, the slope

at any point (x, f(x))

is given by m = 2x

What is the slope

at the point (-1,2)?

m = -2

The equation of the tangent line is

y – 2 = -2(x + 1)

slide7
The limit used to define the slope of a tangent

line is also used to define one of the two funda-

mental operations of calculus --- differentiation

Definition of the Derivative of a Function

f’(x) is read “f prime of x”

Other notations besides f’(x) include:

slide8
Find f’(x) for f(x) = and use the result to find

the slope of the graph of f at the points (1,1) & (4,2). What happens at the point (0,0)?

1

slide9
Therefore, at the point (1,1), the

slope is ½, and at the point (4,2),

the slope is ¼.

What happens at the point (0,0)?

The slope is undefined, since it produces division

by zero.

1 2 3 4

slide11
Theorem 3.1 Alternate Form of the Derivative

The derivative of f at x = c is given by

(x, f(x))

(c, f(c))

c

x

slide12
Derivative from the left and from the right.

Example of a point that is not differentiable.

is continuous at x = 2 but let’s

look at it’s one sided limits.

-1

1

slide13
The 1-sided limits are not equal.

, x is not differentiable at x = 2. Also, the

graph of f does not have a tangent line at the

point (2, 0).

A function is not differentiable at a point at

which its graph has a sharp turn or a vertical

tangent line(y = x1/3 or y = absolute value of x).

Differentiability can also be destroyed by

a discontinuity ( y = the greatest integer of x).

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