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Kathryn
Bollinger, April 14, 2014
Concepts to Know #3
Math 142
6.1-6.7, 8.1, 8.2
• 6.1 - Antiderivatives
Function
General Antiderivative
kf (x)
kF (x) + C
f (x) ± g(x)
F (x) ± G(x) + C
xn
xn+1
+C
n+1
(n 6= −1)
• 6.3/6.4 - Estimating Distance Traveled and
The Definite Integral
Be able to find the area “under a curve” geometrically (using areas of triangles, rectangles, and
circles)
Approximating Area “Under” Curves on [a, b]:
P
Riemann Sums ni=1 f (x∗i )∆x
1. Width of Rectangles (step size) ∆x = b−a
n
(n=number of rect.)
2. Height of Rectangles = function values
(f (x∗i ))
3. x∗i is any point in each subinterval: can be
all left endpoints, all right endpoints, all
midpoints, etc.
Actual Area “Under” Curves: let the number of
rectangles in the Riemann Sum approach
infinity
=⇒ A = lim
n→∞
n
X
f (x∗i )∆x
i=1
1
x
ln |x| + C
ex
Definite Integral:
Z
ex + C
Indefinite
Z Integral = the general antiderivative
=
f ′ (x)dx = f (x) + C
Be able to find a specific antiderivative (value of
“C”), using given info. about the antideriv.
b
f (x)dx = lim
n→∞
a
n
X
f (x∗i )∆x
i=1
= the net area between f (x) and the x-axis
from x = a to x = b
(area above x-axis counted as pos. and area
below x-axis counted as neg.)
Know how to evaluate a definite integral by finding
net area under a graph using geometry or given
area values.
For continuous functions where a ≤ b,
1. If f (x) ≥ 0 for a ≤ x ≤ b, then
Z
b
f (x)dx ≥ 0.
a
2. If f (x) ≥ g(x) for a ≤ x ≤ b, then
• 6.2 - Substitution
Know how and when to use u-substitution to evaluate indefinite and definite integrals.
If using u-substitution with a definite integral,
know the proper notation and how to rewrite
the integral completely in terms of u
(including the limits of integration).
Z
b
a
f (x)dx ≥
Z
b
g(x)dx.
a
3. If m ≤ f (x) ≤ M for a ≤ x ≤ b, then
m(b − a) ≤
Z
b
f (x)dx ≤ M (b − a)
a
where
m = the absolute min of f (x) on [a, b]
and
M = the absolute max of f (x) on [a, b]
(Lower and Upper Bounds for the value of
a definite integral.)
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Kathryn
Bollinger, April 14, 2014
• 6.5 - The Fundamental Theorem of Calculus
Know the properties of definite integrals:
Know that the area between f (x) and g(x) over
[a, b] is given by
For continuous functions f and g:
1.
Z
2.
Z
• 6.6 - Area Between Two Curves
Z
a
f (x) dx = 0
a
Z
b
f (x) dx = −
3.
Z
b
kf (x) dx = k
a
f (x) dx
when f (x) ≥ g(x).
Z
Always draw a picture of the curves and be able to
shade the area...know which curve is the ”top”
curve and which is the ”bottom” curve
=⇒ Start a new integral when the ”top” curve
changes
b
f (x) dx,
a
a
where k is a constant
4.
Z
b
[f (x) ± g(x)] dx =
Zab
f (x) dx ±
5.
Area is always postive.
b
g(x) dx
a
a
Z
Z
Z
b
f (x) dx =
c
f (x) dx +
a
a
Z
b
f (x) dx
c
Suppose f is continuous on [a, b],
Z
b
f (x)dx = F (b) − F (a), where F is any ana
tiderivative of f , that is, F ′ = f .
Know how to use fnInt to approximate definite
integrals
Z
[f (x) − g(x)] dx
a
b
a
b
b
f (x)dx = fnInt(f (x), x, a, b)
• 6.7 - Consumers’ and Producers’ Surplus
Be able to sketch and indicate on a graph where
CS and PS are located.
If p0 is the price of the item in the marketplace
(and x0 is the number of items bought and
sold at that price)
1. CS =
a
Applications of Definite Integrals
Integrating a rate of change function (derivative) over [a, b] gives the net change in the
original function from a to b.
Know how to determine units of a
definite integral.
Know the average value of a continuous function,
f (x), over [a, b] is given by
1
b−a
Z
b
a
f (x) dx
2. PS =
Z
x0
Z 0x0
[D(x) − p0 ] dx
[p0 − S(x)] dx
0
Be able to find an equilibrium point.
• 8.1 - Functions of Several Variables
Be able to evaluate a function of several variables.
Be able to find the domain of a function of several
variables.
Be able to find a function of several variables.
Be able to plot a point in 3D space.
• 8.2 - Partial Derivatives
Be able to find first-order partial derivatives.
Be able to find second-order partial derivatives.