ma110 exam-aid session

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• The Squeeze Theorem

Continuity• Intermediate Value Theorem

Tangent & Secant Lines

Derivatives• Differentiation Rules• Differentiation & Continuity

FUNCTIONS

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:

Symmetry

Even Function Odd Function

Increasing/Decreasing

Increasing Function Decreasing Function

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

Example

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

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

EXPONENT LAWS

Exponent Laws

yxyx aaa

y

xyx

aaa

xyyx aa )(

xxx baab )(

1)

3)

2)

4)

LOGARITHMIC FUNCTIONS

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

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

Example

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

Example

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

Inverse Functions

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

y=x

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

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:

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)

Example

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

LIMITS

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:

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)

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

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

Example

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

2

2

xxx

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

Example

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

xfx )2(

2)()2(2xx

xfxx

CONTINUITY

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.

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

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

)(

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

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.

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

TANGENT & SECANT LINES

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

Tangent & Secant Lines (ctd.)

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

hafhafm

h

)()(lim0

THE DERIVATIVE

The Derivative

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

if this limit exists.

hafhafaf

h

)()(lim)('0

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

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:

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

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

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,

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.

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

given by us and teacher

ADDITIONAL EXAMPLES