4.5 Graphing
1. Summary of Graphing Tools
(1) Find from y = f (x):
(a) Domain: where is f defined?
(b) x intercept: set y = 0 and solve for x
(c) y intercept: set x = 0 and solve for y
(d) Asympotes
(
5.1 Riemann Sums
5.2 Definite Integrals
1. Definitions and Notation
We assume for this section that f is a continuous function on the interval [a, b] and n is a positive
integer.
5
4.5
4
3.5
3
2.5
2
1
4.4 LHospitals Rule
1. Indeterminate Forms
(1)
0
0
(2)
(3) 0
(4)
(5) 00
(6) 0
(7) 1
2. LHospitals Rule
f (x)
. If this limit has the indeterminate form
xa g(x)
f (x)
f 0 (x)
lim
= lim 0
.
xa g(x)
xa
2.7 & 2.8 Definitions of Tangents, Velocity, and Derivatives
1. Recall Tangents and Velocity
(1) Slopes
(a) The slope of the secant line of y = f (x) through (a, f (a) and (b, f (b)
(b) The slope of t
3.9 Related Rates
1. Beginning Related Rates
Recall: If one quantity, say y, varies according to another quantity, say t, then
dy
the instantaneous rate of change of the quantity y with respect to the
3.4 The Chain Rule
1. Theorem
Theorem 1.1 (Version 1). The derivative of a composite function, f g, is
(f g)0 (x) =
Theorem 1.2 (Version 2). If y is a function of u (y = f (u) and u is a function of
x
3.1 Derivatives of Polynomials and Exponential Functions
1. Formulas
For the following c, a and n represents constant real numbers.
(1)
d
(c) =
dx
Example 1.1. Differentiate f (x) =
31
Example 1.2. Di
4.1 Extrema
1. Definitions of Maximum and Minimum Values
Definitions 1.1. Let c be a number in the domain D of a function f . Then f (c) is the
(1) absolute maximum or global maximum at x = c if f (c)
2.6 Infinite Limits and Limits to Infinity
1. Limits at infinity and Horizontal Asymptotes
Definition 1.1. We say the limit as x approaches infinity is L, written lim f (x) = L, if for
x
some x large
3.5 Implicit Differentiation
1. Derivative of Implicit Functions
Example 1.1. Consider the circle, x2 + y 2 = 4.
(1) Draw the circle and tangent lines at (2, 0), (0, 2), and (1, 3).
(2) Find
dy
dx
if
5.3 The Fundamental Theorem of Calculus
Theorem 1.1 (FTC). Assume f is continuous on [a, b].
(1) Then the function g defined by
Z
x
f (t) dt
g(x) =
a
is differentiable on (a, b) and g 0 (x) = f (x).
(
3.6 Derivative of Logarithms
1. Algebra Review
(1) Definition of the logarithm: loga x = y iff
(2) Change of Base Formula: loga x =
(3) Exponent-Log Formula: x =
(4) Product-Sum Formula: loga (xy) =
(
6.4 Work
(1) The work done when a constant force F is applied to move an object with a
distance d is
W = Fd
(2) Suppose the force applied at position x is given by f (x). Then the work done
to move th
4.3 Graphing and Derivatives
1. Increasing and Decreasing
Definitions 1.1. Let f be a continuous function on the interval (a, b).
(1) We say f is increasing on (a, b) if for any two values, x1 and x2
2.5 Continuity
1. Definition
Definition 1.1. A function , f , is continuous at x = a if
Remark 1.1 (Intuitive idea used in algebra based on graphing). A function, f , is
continuous on the interval (a,
3.3 Derivatives of Trigonometric Functions
1. Formulas
(1)
d
(sin x) =
dx
(2)
d
(cos x) =
dx
(3)
d
(tan x) =
dx
(4)
d
(cot x) =
dx
(5)
d
(sec x) =
dx
(6)
d
(csc x) =
dx
Example 1.1. Find f 0 (x) and f
3.7 Rates of Change
1. Recall
The instantaneous velocity of a particle with position at time t given by s(t)
at time t is
The instantaneous acceleration of a particle with position at time t given b
3.8 Exponential Growth and Decay
1. Recall
Definition 1.1. There are many applications where a function is proportional to its
first derivative. In other words,
dy
= ky.
dt
This differential equation
3.11 Hyperbolic Functions
1. Definitions
sinh x =
ex ex
2
cosh x =
ex + ex
2
tanh x =
sinh x
cosh x
coth x =
1
tanh x
sech x =
1
cosh x
csch x =
1
sinh x
2. Derivatives
d
sinh x = cosh x
dx
d
cosh x =
2.3 Limit Laws
1. Basic Application
Theorem 1.1 (Common Sense Limit Laws). If c is a constant and limits lim f (x) and
xa
lim g(x) exist, then
xa
(1) lim [f (x) + g(x)] =
xa
(2) lim [f (x) g(x)] =
xa
6.3 Volume using Cylindrical Shells
1. Cylindrical Shells Method
Example:
Cylindrical Shells Method:
(1) The volume of each shell is
Vi =
2r
| cfw_z
circumference
f (r) dr
| cfw_z |cfw_z
height
thi
5.4 Indefinite Integrals
Theorem 1.1 (FTC). Assume f is continuous on [a, b].
(1) Then the function g defined by
Z
x
f (t) dt
g(x) =
a
is differentiable on (a, b) and g 0 (x) = f (x).
(2) If F (x) is
6.2 Volume
1. Definition of Volume
If the cross-section of a solid shape lying between x = a and x = b has area given by
A(x) at the plane perpendicular to the x-axis through x, then the volume of the
3.2 Product and Quotient Rules
1. Formulas
(1)
d
[f (x)g(x)] =
dx
d f (x)
(2)
=
dx g(x)
2. Examples
Example 2.1. Find the derivative of f (x) = (x2 + 1)(3x + 4).
1
3.2 Product and Quotient Rules
2
3
E
3.10 Linear Approximations and Differentials
1. Linearization
Recall: The tangent line of y = f (x) at the point (a, f (a) is the line the best
approximates the graph of y = f (x) at that point. This
4.7 Optimization
1. Main Steps
Step 1. Read problem and express all information from the problem mathematically.
Use variables to represent any quantity that changes. Numbers may be used
for quantitie
4.9 Antiderivatives
1. Definitions and Properties
(1) A function F is called an antiderivative of f on an interval I if
F 0 (x) = f (x)
for all x I.
(2) Theorem: If F is an antiderivative of f on an i
5.5 Substitution
Remark 1.1. MASTER this method if you are going on to calculus 2. Much of
Chapter 7 is based on substitution and then, of course, everything that follows relies
on what came before.
S
4.2 Rolles Theorem and the Mean Value Theorem
Theorem 1.1 (Rolles Theorem). Let f be a function satisfying the following properties:
(1) f is continuous on the interval [a, b].
(2) f is differentiable