benginning calculus lecture notes 9 - derivative functions

Beginning Calculus- The First and Second Derivative Functions -

Shahrizal Shamsuddin Norashiqin Mohd Idrus

Department of Mathematics,FSMT - UPSI

(LECTURE SLIDES SERIES)

VillaRINO DoMath, FSMT-UPSI

(D7) The First and Second Derivative Functions 1 / 15

The First Derivative Functions The Second Derivative Functions

Learning Outcomes

Interprete the meaning of first and second derivative functions.

Sketch the graphs of first and second derivative functions.




The First Derivative Functions

If f ′ > 0, then f is increasing.

If f ′ < 0, then f is decreasing.




Example

The graphs of f (x) = x2 and its derivative function f ′ (x) = 2x :

4 2 2 4

4

2

2

4

x

y

The graph of f

4 2 2 4

4

2

2

4

x

y

The graphs of f and f ′




Example

The graphs of f (x) = x3 − 3 and its derivative functionf ′ (x) = 3x2 − 1 :

4 2 2 4

10

5

5

10

x

y

The graph of f

4 2 2 4

4

2

2

4

x

y





Example

The graphs of y =1xand

dydx= − 1

x2:

4 2 2 4

2

1

1

2

x

y

The graph of y

4 2 2 4

2

1

1

2

x

y

The graphs of y anddydx




Example

The graphs of f (x) =√x and its derivative function f ′ (x) =

12√x

:

4 2 2 4

4

2

2

4

x

y

The graph of f

4 2 2 4

4

2

2

4

x

y





Example

The graphs of f (x) =1− x2+ x

and its derivative function

f ′ (x) = − 3

(x + 2)2:

4 2 2 4

4

2

2

4

x

y

The graph of f

4 2 2 4

4

2

2

4

x

y





Example

Sketch the graph of the derivative function of the given function:

4 2 2 4

4

2

2

4

x

y

The graph of f

4 2 2 4

4

2

2

4

x

y





The Second Derivative Function

If f ′′ > 0, then f ′ is increasing. The graph of f is concave up.

If f ′′ < 0, then f ′ is decreasing. The graph of f is concave down.

0>′f 0<′f

0=′f

0<′′f

Concave Down

Concave Up

0<′f 0>′f0>′′f

0=′f

Inflection Point

0=′′f




Example

f (x) = x2, Then, f ′ (x) = 2x and f ′′ (x) = 2

4 2 2 4

5

5

10

x

y f(x)f '(x)

f ''(x)




Example

f (x) =√x , Then, f ′ (x) =

12√x, and f ′′ (x) = − 1

4(√x)3

2 4

4

2

0

2

4

x

y

f(x)f '(x)

f ''(x)




Example

Sketch the second derivative function:

4 2 2 4

4

2

2

4

x

yf(x)

f '(x)




Example

Sketch the second derivative function:

4 2 2 4

3

2

1

1

2

3

x

y f(x)

f '(x)



benginning calculus lecture notes 9 - derivative functions

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