calculus 1.1: review of trig/precal a. lines 1. slope: 2. parallel lines—same slope perpendicular...
TRANSCRIPT
Calculus 1.1: Review of Trig/Precal
A. Lines
1. Slope:
2. Parallel lines—Same slopePerpendicular lines—Slopes are opposite
reciprocals
3. Equations of lines:point-slope form: y – y1 = m(x –
x1) slope-intercept form: y = mx + bstandard form: Ax + By = C
mrise
run
y
x
y y
x x
2 1
2 1
B. Functions
1. Function (from set D to set R)—a rule that assigns a unique element in R to each element in D
2. Domain & Range intervals
3. Symmetry: even function if f(-x) = f(x)odd function if f(-x) = -f(x)
4. Piece-wise functions
5. Composite functions:
f a f b whenever a b( ) ( )
C. Inverse functions:
1. f is one-to-one if <horizontal line test>
2.
3. Graphs of inverse functions are reflections across the line y = x
4. To find an inverse function, solve the equation y = f(x) for x in terms of y, then interchange x and y to write y = f-1(x)
f f x and f f x 1 1
D. Exponential & Logarithmic Functions
1. Exponential function:
2. Logarithmic function:
E. Properties of Logarithms:
f x a x( )
f x xa( ) log
1.log ( ) log loga a aXY X Y
2.log log loga a a
X
YX Y
3.log logar
aX r X
4. : logln
lnChange of base x
x
aa
5. : logremember y x a xay
F. Trigonometry Review
sin
cos
tan
y
rx
ry
x
1. Trig Functions: csc
sec
cot
r
y
r
xx
y
Note arclength s r:
2. Remember: Special Right Triangles!!
3. Trig Graphs:
a. Periodicity:
b. Even/Odd:
c. Variations: y = a sin (bx – c) + d
sin sin .
sin sin cos cos
2b g
bg bg
etc
a amplitude d vertical shift
bperiod
c
bphase shift
2
4. Inverse Trig Functions:
<Range:>
y x sin 1
y x cos 1 0 y
y x tan 1
2 2
y
2 2
y
y x csc 1
y x sec 1
y x cot 10 y
Remember: Keep Calculators in Radian Mode!!
Calculus 1.2: Limit of a Function
A. Definition: Limit:
“The limit of f(x), as x approaches a, equals L”—if we can make f(x) arbitrarily close to L by taking x sufficiently close to a (on either side of a) but not equal to a.
Ex 1: see fig 2 p.71 (Stewart)
x a
f x L
lim ( )
B. One-sided limits:
x a
f x L
lim ( ) (from the left)
x a
f x L
lim ( )
lim ( )x af x L
iff
x a x a
f x L f x
lim ( ) lim ( )
C. Estimating Limits using <calculators>
(from the right)
Note:
D. Limit Laws: (if c is a constant and lim ( )x a
f x
and exist) lim ( )x a
g x
1. Sum Rule: lim[ ( ) ( )] lim ( ) lim ( )x a x a x a
f x g x f x g x
2. Difference Rule: lim[ ( ) ( )] lim ( ) lim ( )x a x a x a
f x g x f x g x
3. Constant Multiple Rule: lim[ ( )] lim ( )x a x a
cf x c f x
4. Product Rule: lim[ ( ) ( )] lim ( ) lim ( )x a x a x a
f x g x f x g x
5. Quotient Rule: lim( )
( )
lim ( )
lim ( )x a
x a
x a
f x
g x
f x
g x
( ( ) )g x 0
6. Power Rule: lim[ ( )] [lim ( )]x a x a
f x n f x n
(n is a positive integer)
7. Root Rule: lim ( ) lim ( )x a
f x f xnn
x a
(n is a positive integer)
E. Direct Substitution Property:If f is a polynomial or a rational
function and a is in the domain of f, then:
lim ( ) ( )x a
f x f a
Calculus 1.3: Limits Involving Infinity
A. Definition: (Let f be a function defined on both sides of a) lim ( )
x af x
means that the values of f(x) can be made arbitrarily large by taking x sufficiently close to a (but not equal to a)
Note:
lim ( )x af x
arb. large negative
B. Definition: The line x = a is a vertical asymptote of the curve y = f(x) if at least one of the following is true:
x a
f x
lim ( )x a
f x
lim ( )
x a
f x
lim ( )x a
f x
lim ( )
x a
f x
lim ( )x a
f x
lim ( )
C. Definition: Let f be a function defined on the interval ( , )a
Then lim ( )x
f x L
means that the values of f(x) can be made arbitrarily close to L by taking x sufficiently large.
Note: lim ( )x
f x L
taking x large neg.
D. The line y = L is called the horizontal asymptote of y = f(x) if either:
lim ( )x
f x L
or lim ( )x
f x L
E. Theorem: if r > 0 is rational, then
limx
rx 1 0 lim
xrx 1 0and
F. Method for finding limits at infinity:
1. Divide top and bottom of rational function by the largest power of x in the denominator
2. Simplify using theorem in E above
Calculus 1.4: Continuity
A. DEF: A function f is continuous at a number a if
lim ( ) ( )x af x f a
(assuming f(a) is defined and lim ( ) )x af x exists
**Remember, this means:
lim ( ) lim ( ) lim ( ) ( )x a x a x af x f x f x f a
B. Types of Discontinuity
1. Removable
2. Infinite
3. Jump
4. Oscillating see fig 2.21 p. 80
<Geometrically, the graph of a continuous function can be drawn without removing your pen from the paper>
C. Continuous Functions
1. A function is continuous from the right at a if
2. A function is continuous on an interval [a,b] if it is continuous at every number on the interval
3. The following are continuous at every number in their domains:
polynomials, rational functions
root functions, trig functions
lim ( ) ( )x a
f x f a
D. Intermediate Value Theorem:
Suppose f is continuous on [a,b] and f(a) < N < f(b)
Then there exists a number c in (a,b) such that f(c) = N
Calculus 1.5: Rates of Change
A. Average Rates of Change
1. Average Rate of Change of a function over an interval – the amount of change divided by the length of the interval
2. Secant Line – a line through 2 points on a curve
f
x
f x f x
x x
( ) ( )2 1
2 1
B. Instantaneous Rates of Change
1. Tangent Lines
The tangent line to y = f(x) at the point P(a,f(a)) is the line through P with slope:
mf x f a
x ax a
lim( ) ( )
mf a h f a
hh
lim
( ) ( )0
or
(if the limit exists)
Calculus 1.6: Derivatives
A. Definitions:
1. Differential Calculus—the study of how one quantity changes in relation to another quantity.
2. The derivative of a function f at a number a:
f a
f a h f a
hh( ) lim
( ) ( )0
(if the limit exists), or
f af x f a
x ax a( ) lim
( ) ( )
B. Interpretation of derivatives
1. The tangent line to y = f(x) at (a,f(a)) is the line through (a,f(a)) whose slope is f ’(a)
2. The derivative f ’(a) is the instantaneous rate of change of y = f(x) with respect to x when x = a
C. The Derivative as a Function
1. Definition of the derivative of f(x) as a function:
f x
f x h f x
hh( ) lim
( ) ( )0
Ex: Find f ‘ (x):
1 3. ( )f x x x
2 1. ( )f x x 3
1
2. ( )f x
x
x
Calculus 1.7: Differentiability
A. Other Notation for Derivatives:
f x ydy
dx
df
dx
d
dxf x Df x D f xx( ) ( ) ( ) ( )
B. DEF: A function f is differentiable at a if f ‘(a) exists. It is differentiable on (a,b) if it is differentiable at every number in (a,b).
Ex 1: y x
C. Cases for f NOT to be differentiable at a:
1. Corner – one-sided derivatives differ
2. Cusp – derivatives approach from one side and from the other
3. Vertical Tangent – derivatives approach either or from both sides
4. Discontinuity – removable, infinite, jump or oscillating
Ex 2: Find all points in the domain where f is not differentiable.
State which case each is:
f x x( ) 2 3
D. Graphs of f ’
1. Sketching f ’ when given the graph of f see Stewart p.135 fig
2
a. p. 106 #22 b. p. 105 #13-16c. p. 106-107 #24-26
2. Sketching f when given the graph of f ’
a. p. 107 #27,28