Math 181 Review problems
Find the limits:
x2 8
1. lim
x 2
4
2 x 2 5x 3
2. lim
x 3
x 3
x 1
3. lim
x 1 x 1
3x 2 1
4. lim 2
x 6 x x 2
2x
5. lim
x 1 x 1
6. The table shows a runners distance from the starting line during the first three
seconds of a race.
a)
Scatter plots and curve fitting on the TI-83/84 calculator
Scatter plots:
Begin by deleting or deselecting any functions you already have in the equation editor Y=.
To plot data from a table, press the STAT key and select 1:EDIT. Press Enter.
Enter your d
Test 1 Topics and Review
Limits: (2.2, 2.3, 2.5)
What happens to the y-value of a function when xa, x, or x ?
The limit does not exist if
y is oscillating (not getting close to one value)
the y-values are different on the left and right sides of a
the
Section 5.2
The definite integral: If f is a continuous function on [a, b], divide [a, b] into n subintervals of
ba
equal width x
. Let x0 a , x1 , x2 , ., xn1 , xn b be the
y=f(x)
n
endpoints of the subintervals. Choose sample points x1 * , ., xn * in
t
Section 3.5
When we look at a function y f x , y is expressed explicitly as a function of x. The formula
tells us how to find a y-value that corresponds to a given x-value.
A relationship given by an equation such as x 2 y 2 4 or xy 1 implies, or implicit
Section 3.1
f x h f x
and we have used this definition to find
h0
h
derivatives of some functions, but when we carry out these calculations, we are often repeating
nearly the same process. This will allow us to develop rules for finding derivatives. Thes
Section 2.5
This section looks at limits that involve the concept of infinity in some way. They are different
in some ways from other limits.
Infinite Limits:
1
. We saw that, on both the right and the
x0 x 2
1
left of 0, the function values are increasin
Math 181 Review solutions (continued)
x)
27. Sketch the graph of a continuous function that satisfies all of the following
conditions:
f 1 3
f 1 1
f 3 2
f 3 0
f x 0 if x 3
f x 0 if x 3
f x 0 if 1 x 1
f x 0 if x 1 or x 1
The first derivative tells us there
Final Exam Topics and Review
Limits: (2.3, 2.5)
What happens to the y-value of a function when xa, x, x , x a , or x a ?
The limit does not exist if
y is oscillating (not getting close to one value)
the y-values are different on the left and right sides
Section 5.1
We now return to the area problem that we discussed briefly at the beginning of the course.
Find the area of the region S under the curve y f x between x = a and x = b.
y
y = f(x)
S
a
b
x
For the area of squares, rectangles, triangles, and oth
Section 4.7
Newtons method for approximating roots (solutions) of equations:
If we want to solve any quadratic equation ax 2 bx c 0 , we can use the quadratic formula,
b b 2 4ac
.
x
2a
There exist more complex formulas to solve equations with degree 3 or
Section 2.7
In the previous examples we were concerned with finding the slope or rate of change, that is, the
derivative, for a specific x-value a. We found the derivative of f x x 2 4 x 3 at a = 1,
f 1 2 . However, we will often be interested in finding
Section 5.3 (continued)
The integral of a rate of change of a function gives the net change in the function. For example,
the definite integral of velocity equals the displacement of the moving object, the difference
between the original and final positio
Section 2.2
In the preview of calculus, we looked briefly at a few questions that were important to the
development of calculus. We will come back to look more closely at the problem of finding a
tangent line in section 2.1, but first well look at the ide
Section 4.8
A function F is called an antiderivative of f on an interval I if F x f x for all x in I.
Examples:
F x x3 2 x 2 5x is an antiderivative of f x 3x 2 4 x 5 since F x f x .
F x x3 2 x 2 5x 1 is also an antiderivative of f x 3x 2 4 x 5 since F x
Section 2.6
Now we will generalize what we have said about tangent lines and give a precise definition of
what a tangent line is, so that we can use limit laws to easily find the slope of any tangent line.
As we have seen, the slope of a secant line PQ is
Section 3.8
Remember the general discussion of rates of change from section 2.6.
y f x If x changes from x1 to x2 , x x2 x1 (an increment in x.)
The corresponding change in y is y f x2 f x1 (an increment in y.)
y f x2 f x1
The difference quotient,
= the
Section 3.4 with 1.7
We looked earlier at composite functions, in particular, how to recognize a composite function
and identify the functions forming the composite.
Examples:
F x x 2 1
This is a composite f g f g x , with the inner function g x x 2 1 and
Section 3.9
A curve is close to its tangent line near the point of tangency. If its easy to calculate f a for a
particular value of a, but difficult for nearby values, we can use the tangent to approximate the
function near a.
For the function y f x , the
Section 1.6
This section is a review of inverse functions. We will want to be familiar with inverses,
particularly for logarithmic functions and for inverse trigonometric functions.
If y f x is a one-to-one function (each y-value in the range corresponds
Section 2.3
In this section we look at ways to accurately compute limits of functions, without having the
possible problems involved in using a table of values or a graph. The following limit laws are
properties that we can use to find limits of a variety
Section 3.3 and Trig Review
In this section, we look at trig functions and their derivatives, so we will review a little bit; if you
need more review, you can look at Appendix C on page A17 in the back of the textbook. There
is also a summary of important
Section 4.6
In this section we look at applications of maximum and minimum values. Optimization is the
process of finding the best value for a function. We may want to maximize the volume of a
container, or to minimize the distance to be traveled or the c
Test 3 Topics and Review
Applications of the derivative:
Related rates: (4.1)
Solve for an unknown rate of change of a variable, given some values of the variables
and rates of change. You can expect the relationship between the variables to be either a
k
Section 4.1
Related Rates:
When the radius of a circle changes over time, the area changes.
When the volume of a sphere changes, the surface area changes.
When the depth of water in a tank changes, the volume changes.
If two variables are related by an eq
Section 3.2
The derivative of a product of functions is not the product of the derivatives.
Example:
f x x , g x x 2
d
The product of the functions, f x g x x3 , which has derivative
f x g x 3x 2
dx
The derivatives of the functions are f x 1 and g x 2 x
Section 1.2
A mathematical model is a mathematical description of a real-world phenomenon. The process
of modeling involves observing a real-world problem, using data or other observations to
formulate equations or functions (the model) describing the pro
Introduction
You should know the basics from precalculus very well: various types of functions and
equations and their graphs, as well as accurate algebraic techniques for simplifying expressions
and solving equations. We will be quickly reviewing some of
Section 4.3
We already discussed in section 2.8 how to sketch a graph if we have information about where
the graph is increasing and decreasing, concave up and concave down, etc. At that time we were
given information about where the first and second deri
Section 5.3
In the last two sections in this class, we look at the two parts of an important theorem, the
Fundamental Theorem of Calculus. First we look at the Evaluation Theorem, which tells us how
to evaluate definite integrals without using the limit o
Notes from 2.7
Calculus I Test #2
Derivatives
1.
Interpreting
a. If a derivative is positive, it tells us that the function (not the derivative as a
function) is increasing; likewise, if the derivative is negative, it tells us that the
function is decrea
Notes (random) from 2.1 2.6
CALCULUS I TEST #1
RATES OF CHANGE
Finding the slope of a tangent line:
1. When finding the slope of a tangent (a, f(a), we first make a secant line which is some
distance h from the point (a, f(a)
-noted as (a+h, f(a+h) as the
Review and Background for Calculus
Differentiable a function is differentiable if it is continuous (i.e. a derivative exists for each point in
the functions domain no breaks, undefined point, discontinuities) AND there are no vertical tangent
lines (i.e.
Section 2.4
Definition: A function f is continuous at a number a if lim f x f a . Otherwise f is
xa
discontinuous at a, f has a discontinuity at a. We can think of continuity as requiring three
things:
1. f a must be defined.
2. lim f x must exist.
xa
3.
Math 181 Review solutions
Find the limits:
x2 8
1. lim
3 (continuous function, direct substitution)
x 2
4
2 x 1x 3 lim 2 x 1 7
2 x 2 5x 3
2. lim
lim
x 3
x 3
x3
x 3
x 3
x 1 lim 1 1
x 1 x 1
x 1
3. lim
lim
lim
x 1 x 1 x 1
x 1 x 1 x 1
x 1 x 1
x 1 x 1
2
3x