I’m going nutsover derivatives!!!

2.1The Derivative and theTangent Line Problem

Calculus grew out of 4 major problems that European mathematicians were working onin the seventeenth century.

1. The tangent line problem

2. The velocity and acceleration problem

3. The minimum and maximum problem

4. The area problem

The tangent line problem

(c, f(c))

secant line

f(c+ ) – f(c)x(c, f(c)) is the point of tangency and is a second point on the graph of f.

)()( cfxcf

x

)(, xcfxc

The slope between these two points is

cxccfxcfm

)()(

sec

xcfxcf

)()(

Definition of Tangent Line with Slope m

mx

cfxcfx

)()(lim0

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)

xxfxxf

x

)()(lim0

x

xxxx

11)(lim22

0

x

xxxxxx

112lim222

0

xxxx

x

)2(lim0

x2

Therefore, the slopeat 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)

f(x) = x2 + 1

The limit used to define the slope of a tangentline is also used to define one of the two funda-mental operations of calculus --- differentiation

Definition of the Derivative of a Function

xxfxxfxf

x

)()(lim)('0

f’(x) is read “f prime of x” Other notations besides f’(x) include:

][)],([,', yDxfdxdy

dxdy

x

Find f’(x) for f(x) = and use the result to findthe slope of the graph of f at the points (1,1) & (4,2). What happens at the point (0,0)?

,x

xxfxxfxf

x

)()(lim)('0

xxxxxf

x

0lim)('

xxxxxx

xxxxxxx

x

)(lim0 xxxx

xxxx

0lim

1

xxxx

1lim0 x2

1

xmxf

21)('

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 divisionby zero.

21

m

41

m

1 2 3 4

Find the derivative with respect to t for the

function .2t

y

ttfttf

dxdy

t

)()(lim0

tttt

t

22

lim0

1

)()(22

lim0 t

tttttt

t

ttttttt

t

1)(222lim

0

2

2t

Theorem 3.1 Alternate Form of the Derivative

The derivative of f at x = c is given by

cxcfxfcf

cx

)()(lim)('

(c, f(c)))()( cfxf

cxx c x

(x, f(x))

Derivative from the left and from the right.

cxcfxf

cx

)()(lim cxcfxf

cx

)()(lim

Example of a point that is not differentiable.

2)( xxf is continuous at x = 2 but let’s look at it’s one sided limits.

2)2()(lim

2 xfxf

x

202

lim2 xx

x-1

2)2()(lim

2 xfxf

x

202

lim2 xx

x1

The 1-sided limits are not equal.

, x is not differentiable at x = 2. Also, thegraph of f does not have a tangent line at the point (2, 0).

A function is not differentiable at a point atwhich its graph has a sharp turn or a verticaltangent line(y = x1/3 or y = absolute value of x). Differentiability can also be destroyed by a discontinuity ( y = the greatest integer of x).