Transcript Review Poster

Review Poster
First Derivative Test [Local min/max]
If x = c is a critical value on f and f’ changes
sign at x = c…
(i) f has a local max at x = c if f’ changes
from >0 to <0
(ii) f has a local min at x = c if f’ changes
from < 0 to > 0
Number Line Analysis [instead of the big
chart we used to
make]
(iii) No sign change at x = c, no local min/max
Second Derivative Test [Also a test for
local min/max, not a test for concavity
or points of inflection]
If x = c is a critical value on f [meaning f’(c) = 0 or
is undefined] and f”(c) exists…
(i) If f”(c) > 0, x = c is a local min on f
(ii) If f”(c) < 0, x = c is a local max on f
(iii) If f”(c) = 0, then the test fails and we don’t
know anything.
Test for Concavity on f [Points of
Inflection]
Evaluate f” at points where f’=0
•Point of inflection on f at x = c, f changes from
concave down to concave up
• Point of inflection on f at x = c, f changes from
concave up to concave down
Linear Approximation
Use equation of a line tangent to f at a point
(x, f(x)) to estimate values of f(x) close to the
point of tangency
1) If f is concave up (f” > 0), then the linear approximation will
be less than the true value
2) If f is concave down (f”< 0), then the linear approximation
will be greater than the true value.
Properties of f(x) = ex
• Inverse is y = lnx
• Domain (-∞,∞)
• Range (0, ∞)
•
• elnx = x
• ln ex = x
•
• ea eb = ea+b
•
•
Properties of f(x) = lnx
•
•
•
•
Inverse is y = ex
Domain (0, ∞)
Range (- ∞, ∞)
Always concave
down
• Reflection over x-axis: -lnx
• reflect over y-axis: ln(-x)
• ln(ab) = lna + lnb
• ln (a/b) = lna - lnb
• ln(ak) = klna
• ln x <0 if 0<x<1
• horizon. shift a units
left: ln(x + a)
• vert. shift a units
• horizon. shift a units
right: ln(x – a)
• vert. shift a units
up: ln(x) + a
down: ln (x) - a
Chain Rule (Composite Functions)
Rules for Differentiation
Product rule:
Quotient rule:
Implicit Differentiation
When differentiating with respect to x (or t or θ)
1) Differentiate both sides with respect to x, t,
or θ.
2) Collect all terms with on one side of the
equation.
3) Factor out .
4) Solve for .
An Example of Implicit Differentiation
if 2xy + y3+ x2 = 7
Find
• 2y + 2x
+ 3y2
+ 2x = 0
•
(2x + 3y2)=-2x - 2y
•
=
Another (slightly different) example of
implicit differentiation
If x2 + y2 = 10, find
.
I will pause here to let you catch up on copying
and try to solve this problem on your own.
2x + 2y
=0
= -x/y
=
=
=
= (-y2 – x3 )/y3
Line Tangent to Curve at a Point
- Need slope (derivative) at a point (original
function)
- A line normal to a curve at a point is
______________ to the tangent line at that
point. (The slopes of these lines will be
___________ _____________)
Related Rates (The rates of change of
two items are dependent)
1. Sketch
2. Identify what you know and what you want
to find.
3. Write an equation.
4. Take the derivative of both sides of the
equation.
5. Solve.
Big Section: Integrals
-
Approximate area under a curve
Riemann Sum =
Left endpoint
Right endpoint
Midpoint
Inscribed and Circumscribed
Rectangles
Inscribed Rectangles
- underestimate
- happens when f is decreasing and you use
right end point OR when f is increasing and
you use left endpoint
• Circumscribed Rectangles
-overestimate
- happens when f is decreasing and you
use left end point OR when f is increasing
and you use right endpoint
Trapezoidal Rule – most accurate
approximation
If f is continuous on [a,b] 
•As n  ∞, this estimate is extremely accurate
• Trapezoidal rule is always the average of left
and right Riemann sums
Fundamental Theorem of Calculus
Part I
If F’ = f,
Fundamental Theorem of Calculus
Part II
where a is a constant and x is a
function
Properties of Definite Integrals
•
•
•
•
Oh man. This is taking me so long to
type. So. Long.
• If f is an odd function (symmetric about the
origin, (a,b) (-a,-b)) , then
• If f is an even function (symmetric about the
y-axis, (a,b) (-a,b))), then
• If f(x) ≥ g(x) on [a,b] then
Definition of Definite Integral
Average Value of a Function on [a,b]
• M(x) = Average Value =
•
• So, (b-a)(Avg. Val.)=
Total Distance Vs. Net Distance
Net Distance over time [a,b] =
Total Distance over time [a,b] =
Area Between Two Curves
If f(x) and g(x) are continuous on [a,b] such that
f(x) g(x), then the area between f(x) and g(x)
is given by
Area =
Volumes
There are basically three types of volume problems…
1. Volume by Rotation – Disc - (x-section is a circle)
(where R is the radius
from the axis of rev.)
2. Volume by Rotation – Washer -(x-section is a circle)
R(x) = radius from axis of revolution to
outer figure
r(x) = radius from axis of revolution to
inner figure
3. Volume of a Known Cross-Section
(foam projects)
V=
A(x) is the area of a known
cross-section
You’re almost there! Only two
more slides! (After this one)
Next Big Section: Differential
Equations and Slope Fields
Differential Equations
1. Separate dy and dx algebraically.
[Separation of variables.]
2.
both sides. This will create a c value.
The general solution has a c in it.
3. Solve for c using the initial conditions.
Use this c value to write the particular
solution.
Slope Fields
-Show a graphical solution to differential equations
- Big picture made of tangent segments is the solution
- The slope of each individual tangent segment is the
value of
at that point
The
End