DERIVATIVES

Derivatives: Slope of a linear function

How steep is the graph of f: y = 2x 1?

The slopem = 2

measures how steep the

line is.

Derivatives: Slope of a quadratic function

How steep is the graph of f: y = x2 6x + 8?

Differs from point to point! Slope of the

TANGENT in a point measures how steep the graph is in that

point! Name:

DERIVATIVE!

Derivatives: Slope of a general function

We DEFINE:

The slope of (the tangent line to) the graph of f at the point with first coordinate a

=the DERIVATIVE OF THE FUNCTION f at a,

denoted f’(a)

Exercise 1 (c), (b), (a)

Figuur

Derivatives: Slope of a general function

Derivatives: as a FUNCTION

Derivative differs from point to point and is therefore itself a FUNCTION! (http://www.geogebra.org/en/examples/function_slope/function_slope1.html)

Notation: f ’

Calculation of the derivative at a point

STEP 1: equation of the derivative FUNCTION using rules proven by mathematicians.

Rules: r, s, a, b, c numbers

0r 1x

1r rx rx

r s r sax bx c a x b x c

Example:

3

3

2

2

5 2 8

5 2 8

5 3 2 1 0

15 2

x x

x x

x

x

STEP 2: plug in the x-coordinate in the equation of the derivative FUNCTION

Example:Example: if f(x) = 5x3 + 2x + 8, what is the slope of the graph of f at the point with x-coordinate 2?

in step 1 we found: f ’(x) = 15x2 +2hence: f ’(2) = 15 22 + 2 = 62

this slope is given by f ’(2)!

Calculation of the derivative at a point

Derivatives: Exercises

• Exercise 2 (e), (c)• Exercise 3 (b)• Exercise 4

Figure• Exercise 6

FigureSupplementary exercises• rest of exercises 2 and 3• Exercise 7

Derivatives: Meaning in several contexts

Taxi ride company A: y = 2x + 5.• x: length of the ride (in km), y: price, cost of the ride• q = 5: base price• m = 2: price per km, also MARGINAL COST

(constant!)

If f(x) = 2x + 5 then f ’(x) = (2x + 5)’ = 2(x)’ + (5)’ = 2 and hence the DERIVATIVE IS EQUAL TO THE MARGINAL COST. (constant!)

This holds in general: if TC = f(q) (linear or not linear!) then MK = f ’(q)

Exercises 5 en 8

Derivatives: Summary

• Derivative = slope of a tangent• Derivative is a function itself• Calculation at a point - rules to calculate derivative function - plug in x-coordinate of point• Meaning in several contexts: e.g. marginal cost

Exercise 1 (3)

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Exercise 1 (2)

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Exercise 1 (1)

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Exercise 4

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1 1

4

3

2

1

1

2

3

4

X

Y

Exercise 6

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3 2 1 1 2

5

5

X

Y