DERIVATIVES Derivatives: Slope of a linear function How steep is the graph of f: y = 2x  1? The slope m = 2 measures how steep the line is.

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Presentation transcript:

DERIVATIVES

Derivatives: Slope of a linear function How steep is the graph of f: y = 2x  1? The slope m = 2 measures how steep the line is.

Derivatives: Slope of a quadratic function How steep is the graph of f: y = x 2  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! Derivative differs from point to point and is therefore itself a FUNCTION! ( 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 Example:

STEP 2: plug in the x-coordinate in the equation of the derivative FUNCTION Example:if f(x) = 5x 3 + 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) = 15x 2 +2 hence: f ’(2) = 15  = 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 Figure Supplementary 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) Back

Exercise 1 (2) Back

Exercise 1 (1) Back

Exercise 4 Back

Exercise 6 Back