ma110 exam-aid session
DESCRIPTION
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 PresentationTRANSCRIPT
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