Differentiation Lesson 1 PDF

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DIFFERENTIATION y

DEF: Derivative of a curve at point A is the slope of the tangent to the curve at that point

C B A

x y

a

c

Tan d = d b

x

y B f(a+h)

f

f(a+h) — f(a) f(a)

DEF: Derivative of f(x) at a point is:

A

REMEMBER: a

a+h h

x

THE EQUATION OF THE TANGENT OF f at A is:

EXAMPLE: COMPUTE f’(a) when f(a) = x² for a=-1 and a=½

f’(½) = 1

f’(-1) = -2

y

-1

-½

½

x

1

EXAMPLE: SHOW THAT f’(x) = -1/x² from f(x) = 1/x

I

I

INCREASING IN I

STRICTLY INCREASING IN I

If f(x2) ≥ f(x1) whenever x2 > x1 then f is INCREASING in I

If f(x2) > f(x1) whenever x2 > x1 then f is STRICTLY INCREASING in I

I

I

DECREASING IN I

STRICTLY DECREASING IN I

If f(x2) ≤ f(x1) whenever x2 > x1 then f is DECREASING in I

If f(x2) < f(x1) whenever x2 > x1 then f is STRICTLY DECREASING in I

If f’(x) ≥ 0 for all x in the interval I → f is INCREASING in I If f’(x) ≤ 0 for all x in the interval I → f is DECREASING in I If f’(x) = 0 for all x in the interval I → f is CONSTANT in I

#

DECREASING in [ - ∞, 0]

#

INCREASING in [0, ∞]

+ 3 1 — 1

Therefore: #

#

#

#

#

• INCREASING in [1, 3] • DECREASING in [ - ∞, 1] and [3, ∞]

3 —

DEF: the instantaneous rate of change of f at a is:

DEF: the relative rate of change of f at a is:

C(x) is the COST of producing x units R(x) is the REVENUE from selling x units π(x) is R(x) — C(x) = PROFIT C’(x) = MARGINAL COST R’(x) = MARGINAL REVENUE π’(x) = MARGINAL PROFIT

When h is “small”:

If h = 1:

MARGINAL COST IS APPROXIMATELY EQUAL THE INCREMENTAL COST OF PRODUCING AN EXTRA UNIT...