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MA110 Exam-AID Session By: Riley Furoy & Amanda Nichol

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MA110 Exam-AID Session . By: Riley Furoy & Amanda Nichol. Agenda. Functions Symmetry Increasing/Decreasing Composite Inverse Exponent Laws Logarithmic Functions Laws of Logarithms Change of Base Formula The Natural Logarithm Limits Limit Laws Horizontal & Vertical Asymptotes - PowerPoint PPT Presentation

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Page 1: MA110 Exam-AID Session

MA110 Exam-AID Session By: Riley Furoy & Amanda Nichol

Page 2: MA110 Exam-AID Session

Agenda

Functions• Symmetry• Increasing/Decreasing• Composite• Inverse

Exponent Laws

Logarithmic Functions• Laws of Logarithms• Change of Base Formula • The Natural Logarithm

Limits

• Limit Laws• Horizontal & Vertical Asymptotes• The Squeeze Theorem

Continuity• Intermediate Value Theorem

Tangent & Secant Lines

Derivatives• Differentiation Rules• Differentiation & Continuity

Page 3: MA110 Exam-AID Session

FUNCTIONS

Page 4: MA110 Exam-AID Session

Functions

x1x2x3...

f(x1)f(x2)f(x3)...

D E

{x1,x2,…} is the domain {f(x1),f(x2),… } is the range

A function f:

Page 5: MA110 Exam-AID Session

Symmetry

Even Function Odd Function

Page 6: MA110 Exam-AID Session

Increasing/Decreasing

Increasing Function Decreasing Function

Page 7: MA110 Exam-AID Session

Composite Functions

x1x2x3...

g(x1)g(x2)g(x3)...

{x1,x2,…} is the domain of g

{g(x1), g(x2),… } is the range of g; a subset of

this is the domain of f ∘ g

A function f(g(x)):

f(g(x1))f(g(x2))f(g(x3))...

{f(g(x1)), f(g(x2)),… } is the range of f ∘ g

Page 8: MA110 Exam-AID Session

Example

e.g. Find the domain of the function f∘ g given:

xf(x)= and 2log2 xg(x)=

Page 9: MA110 Exam-AID Session

EXPONENT LAWS

Page 10: MA110 Exam-AID Session

Exponent Laws

yxyx aaa

y

xyx

aaa

xyyx aa )(

xxx baab )(

1)

3)

2)

4)

Page 11: MA110 Exam-AID Session

LOGARITHMIC FUNCTIONS

Page 12: MA110 Exam-AID Session

Logarithmic Functions

logax = y ⇔ ay = x

Cancellation Equations:  for all x RЄ 

for all x > 0

xa xa )(log

xa xa log

Laws of Logarithms:

1)

2)

3)

yxxy aaa loglog)(log

yxyx

aaa loglog)(log

xrx ar

a log)(log

Page 13: MA110 Exam-AID Session

Logarithmic Functions (ctd.)

In the special case where a = e, we have the natural logarithm:

Cancellation Equations:  for all x RЄ 

for all x > 0

xe x )ln(

xe x ln

Change of Base Formula:

axx

b

ba log

log)(log

xexx ye lnlog

Page 14: MA110 Exam-AID Session

Example

e.g. Solve )1ln(1ln xx

Page 15: MA110 Exam-AID Session

Example

e.g. Solve )4(log)2(log 42 xx

Page 16: MA110 Exam-AID Session

Inverse Functions

A One-to-One Function… …and its Inverse Function

y=x

The inverse is essentially a reflection along the line y=x.

Page 17: MA110 Exam-AID Session

Inverse Functions (ctd.)

Cancellation Equations:

f -1(f(x)) = x for all x in Af(f -1(x)) = x for all x in B

)(1)(1xf

xf

Note:

Page 18: MA110 Exam-AID Session

Inverse Functions (ctd.)

Steps to Finding the Inverse:

Step 1 – Write y = f(x)

Step 2 – Interchange x and y

Step 3 – Solve this equation for x in terms of y

The resulting equation is y = f -1(x)

Page 19: MA110 Exam-AID Session

Example

e.g. Find the inverse of 1)( xexf

Page 20: MA110 Exam-AID Session

LIMITS

Page 21: MA110 Exam-AID Session

The Limit of a FunctionLet f(x) be defined for all x in an open interval containing the number a (except possibly at a itself). Then we write

Lxfax

)(lim

If f(x) can be made arbitrarily close to L whenever x is sufficiently close (but not equal to) a.

)(lim)(lim xfLxfaxax

In order for the limit to exist at a:

Page 22: MA110 Exam-AID Session

Limit Laws

)(lim)(lim)]()([lim xgxfxgxfaxaxax

)(lim)(lim)]()([lim xgxfxgxfaxaxax

)(lim)]([lim xfcxfcaxax

)(lim)(lim)]()([lim xgxfxgxfaxaxax

)(lim

)(lim

)()(lim

xg

xf

xgxf

ax

ax

ax

Addition Law

Subtraction Law

Constant Law

Multiplication Law

Division Law(Holds only if the bottom limit

is not zero)

Page 23: MA110 Exam-AID Session

Limit Laws (ctd.)

n

ax

n

axxfxf )](lim[)]([lim

ccax

lim

axax

lim

nn

axax

lim

nn

axax

lim

nax

nax

xfxf )(lim)(lim

Power Law

Root Law

Page 24: MA110 Exam-AID Session

AsymptotesThe line x = a is a vertical asymptote of y = f(x) if at least one of the following conditions is true:

The line y = a is a horizontal asymptote of y = f(x) if at least one of the following conditions is true:

)(lim xfax

)(lim xfax

)(lim xfax

axfx

)(lim axfx

)(lim

Page 25: MA110 Exam-AID Session

Example

e.g. Determine if has any vertical or horizontal asymptotes.9413

2

2

xxx

Page 26: MA110 Exam-AID Session

The Squeeze Theorem

If f(x) ≤ g(x) ≤ h(x) when x is near a (except possibly at a) and

Lxhxfaxax

)(lim)(lim

then:

Lxgax

)(lim

Page 27: MA110 Exam-AID Session

Example

e.g. Find given for all x (0, 2). ∈)(lim1

xfx )2(

2)()2(2xx

xfxx

Page 28: MA110 Exam-AID Session

CONTINUITY

Page 29: MA110 Exam-AID Session

ContinuityA function f is continuous at a if

)()(lim afxfax

This implies that 3 conditions must be met:

1) f(a) is defined

2) exists

3)

)(lim xfax

)()(lim afxfax

A function is continuous on an interval if it is continuous at every point on the interval.

Page 30: MA110 Exam-AID Session

Continuity (ctd.)

Any polynomial function is continuous everywhere; that is, it is continuous on . Similarly, any rational function is continuous wherever it is defined; that it, it is continuous on its domain.

),(

If f and g are continuous at a, and c is a constant, then the following functions are also continuous at a:

1) f + g

2) f – g

3) c ∙ f

4) f ∙ g

5) If g(a) ≠ 0 gf

Page 31: MA110 Exam-AID Session

Example

e.g. Let .

Determine the value(s) of x at which f is undefined and use your answer to determine the domain of f.

x

x

eexf21

)(

Page 32: MA110 Exam-AID Session

Example

e.g. Given

Determine the value of the constant c for which f is continuous at x = 2.

)(xf xcx 22 cxx 3

if x < 2if x ≥ 2

Page 33: MA110 Exam-AID Session

Intermediate Value Theorem

Suppose that f is continuous on the closed interval (a,b) and let N be any number between f(a) and f(b), where f(a) ≠ f(b). Then there exists a number c in (a,b) such that f(c) = N.

Page 34: MA110 Exam-AID Session

Example

e.g. Use the Intermediate Value Theorem to show that the equation

(where x > 0) has at least one solution on the interval (1,e).

xex ln

Page 35: MA110 Exam-AID Session

TANGENT & SECANT LINES

Page 36: MA110 Exam-AID Session

Tangent & Secant Lines

Taking points P(a, f(a)) and Q(x, f(x)) for any function, the slope of the line between these two points (the secant) is

If we take (and make Q closer and closer to P), the slope

of the secant line converges to the slope of the tangent line at P.

axafxfm

)()(

axafxf

ax

)()(lim

Page 37: MA110 Exam-AID Session

Tangent & Secant Lines (ctd.)

The tangent measures the instantaneous rate of change of the function, and its slope is

hafhafm

h

)()(lim0

Page 38: MA110 Exam-AID Session

THE DERIVATIVE

Page 39: MA110 Exam-AID Session

The Derivative

The derivative of a function f at a number a, denoted f ’(a), is

if this limit exists.

hafhafaf

h

)()(lim)('0

Page 40: MA110 Exam-AID Session

Example

e.g. Use the limit definition to find the slope of the tangent line l to the curve

at the point .

14)(

x

xxf

38,2P

Page 41: MA110 Exam-AID Session

Differentiation Rules

The method for finding the derivative outlined above, known as “first principles”, can be tedious for complicated functions. Therefore there are some shortcuts that we use:

1xdxd 0c

dxd

1 nn xnxdxd

)(')( xfcxfcdxd

Power Rule:

Constant Rule:

Page 42: MA110 Exam-AID Session

Differentiation Rules (ctd.)

)](')([)]()('[)]()([ xgxfxgxfxgxfdxd

2)]([)](')([)]()('[

)()(

xgxgxfxgxf

xgxf

dxd

Exponential Differentiation:

Logarithmic Differentiation:

Quotient Rule:

Chain Rule:

Product Rule:

)('))(('))(( xgxgfxgfdxd

aaxfadxd xfxf ln)(' )()(

)()(')(ln

xfxfxf

dxd

Page 43: MA110 Exam-AID Session

Example

We can use the derivative laws to solve the previous example. To do so, find the derivative of f and substitute in the x-coordinate of the point P.

e.g. Find the slope of the tangent line l to the curve

at the point .

14)(

x

xxf

38,2P

Page 44: MA110 Exam-AID Session

Differentiation & Continuity

If a function is differentiable at a, then it is continuous at a.

Note that the converse of this theorem is NOT true.

Which of the above functions are differentiable at a?

For example,

Page 45: MA110 Exam-AID Session

Differentiation & Continuity

Answer: NONE

The first graph has a “cusp”, and so the slope of the tangent is different on either side of a.

The second graph is not continuous, and therefore cannot be differentiable.

The third graph has a vertical tangent, so the derivative does not exist at a.

Page 46: MA110 Exam-AID Session

Study Tips• Practice, Practice, Practice• Go over past tests and all examples

given by us and teacher

Page 47: MA110 Exam-AID Session

ADDITIONAL EXAMPLES