Name: _ Class: _ Date: _
Math2413 Test1 review
1.
Let f (x ) =
4 x,
0,
x1
.
x=1
Determine the following limit. (Hint: Use the graph to calculate the limit.)
lim f (x )
x1
2. Determine the following limit. (Hint: Use the graph to calculate the limit.)
li
4.1.1
4.1: Antiderivatives and Indefinite Integration
Definition: An antiderivative of f is a function whose derivative is f.
i.e. A function F is an antiderivative of f if F '( x)
Example 1:
f ( x) .
_ is an antiderivative of f ( x) 3 x 2 5 .
What are so
4.3.1
4.3: Riemann Sums and Definite Integrals
Riemann sums:
So far we have discussed upper sums, lower sums, right endpoint sums, left endpoint sums, and
midpoint sums, all used to approximate the area under the graph of a nonnegative continuous
function
4.4.1
4.4: The Fundamental Theorem of Calculus
Evaluating the area under a curve by calculating the areas of rectangles, adding them up, and
letting taking the limit as n
is okay in theory but is tedious at best and not very practical.
Fortunately, there
5.4.1
5.4: Exponential Functions: Differentiation and Integration
Short Review:
An exponential function takes the form f ( x) b x , where b 0 and b 1 .
For any exponential function f ( x) b x , the graph looks like one of the following.
b 1
0 b 1
Notice:
2.6.1
2.6: Related rates
General idea for solving rate problems:
1. Draw a sketch if applicable. The only dimensions you put on your sketch should be those that do
not change.
2. Write down, in calculus notation, the rates you know and want.
3. Write an e
5.2.1
5.2: The Natural Logarithmic Function: Integration
Using the derivative of the natural logarithmic function to obtain an antiderivative:
Example 1:
Find the derivative of g ( x ) ln x .
Note that f ( x) ln x has the same derivative as g ( x) ln x .
3.6.1
3.6: A Summary of Curve Sketching
Steps for Curve Sketching
1. Determine the domain of f.
2. Find the x-intercepts and y-intercept, if any.
3. Determine the end behavior of f, that is, the behavior for large values of x (limits at
4.
5.
6.
7.
8.
9.
4.1.1
4.1: Antiderivatives and Indefinite Integration
Definition: An antiderivative of f is a function whose derivative is f.
i.e. A function F is an antiderivative of f if F '( x)
Example 1:
f ( x) .
_ is an antiderivative of f ( x) 3 x 2 5 .
What are so
3.1.1
3.1: Extrema on an Interval
Absolute maximum and minimum:
If f ( x)
of f.
f (c) for every x in the domain of f, then f (c) is the maximum, or absolute maximum,
If f ( x) f (c ) for every x in the domain of f, then f (c) is the minimum, or absolute m
3.3.1
3.3: Increasing and Decreasing Functions and the First Derivative Test
Increasing and decreasing functions:
A function f is said to be increasing on the interval (a, b) if, for any two numbers x1 and x2 in ( a, b) ,
f ( x1 ) f ( x2 ) whenever x1
is
4.5.1
4.5: Integration by Substitution
Most functions cannot be integrated using only the formulas we have learned so far. In
Calculus II, you will learn several advanced integration techniques. For now, well learn just one
new integration technique, call
3.7.1
3.7: Optimization Problems
We often need to solve problems involving optimization: finding the maximum or minimum of some
quantity.
Process for Solving Optimization Problems
1. Assign a variable to each quantity mentioned. If possible, draw and labe
3.9.1
3.9: Differentials
Differentials:
If y
f ( x) is a differentiable function, we can let dx represent an amount of change in x.
f '( x)dx .
Then the differential dy is defined to be dy
dy is an approximation to
y
f (x
x)
f ( x) , which is the actual c
4.2.1
4.2: Area
There are two fundamental problems addressed by calculus:
1. Finding the slope of a tangent line to a function at a particular location (differential
calculus).
2. Finding the area between the graph of a function and the x-axis over an int
5.2.1
5.2: The Natural Logarithmic Function: Integration
Using the derivative of the natural logarithmic function to obtain an antiderivative:
Example 1:
Find the derivative of g ( x ) ln x .
Note that f ( x) ln x has the same derivative as g ( x) ln x .
5.1.1
5.1: The Natural Logarithmic Function: Differentiation
An algebraic approach to logarithms:
Definition: log b x
y is equivalent to b y
x.
The functions f ( x) b x and g ( x) log b x are inverses of each other.
b is called the base of the logarithm.
5.6.1
5.6: Inverse Trigonometric Functions Differentiation
Because none of the trigonometric functions are one-to-one, none of them have an inverse
function. To overcome this problem, the domain of each function is restricted so as to produce a
one-to-one
5.1.1
5.1: The Natural Logarithmic Function: Differentiation
An algebraic approach to logarithms:
Definition: log b x
y is equivalent to b y
x.
The functions f ( x) b x and g ( x) log b x are inverses of each other.
b is called the base of the logarithm.
3.7.1
3.7: Optimization Problems
We often need to solve problems involving optimization: finding the maximum or minimum of some
quantity.
Process for Solving Optimization Problems
1. Assign a variable to each quantity mentioned. If possible, draw and labe
Name: _ Class: _ Date: _
2413final review
Short Answer
1. Find the limit.
lim
x4
x+5
x1
2. Find the limit (if it exists).
lim
x5
x+4 3
x5
3. Find the lmit.
x
lim tan
3
x
1
ID: A
Name: _
ID: A
4. Determine the following limit. (Hint: Use the grap
Math2413 Calculus1
Test2 Review
1. Find the derivative of the function.
7
f (t ) = (8 + 2t ) 5
2. Find the derivative of the function.
f (x ) = x 5
8 7x
3. Find the derivative of the function.
x +2 6
g (x ) = 2
x +5
4. Find the derivative of the functio
Math2413 Test3 review
1. Find the indefinite integral
13
x 3 dx.
2. Solve the differential equation.
df
12z + 3
=
2
dz 2
6z + 3z + 5
6
3. Find the indefinite integral u 2 4 + u 3 du .
4. Find the indefinite integral of the following function.
cosz
sin5
1.4.1
1.4: Continuity and One-Sided Limits
One-Sided Limits:
lim f ( x )
x
lim f ( x)
x
L means that f ( x) approaches L as x approaches a from the right.
a
lim f ( x)
x
L means that f ( x) approaches L as x approaches a from the left.
a
a
Example 1:
L if
2.5.1
2.5: Implicit Differentiation
Example 1:
Given the equation x3 4 y 9 x 2
5 , find
dy
by
dx
a) Solving explicitly for y.
b) Implicit differentiation.
Example 2:
Find
dy
for xy
dx
Example 3:
Find
dy
for the equation x3 y 2 x 2 y 3
dx
4.
x2 3 0 .
2.5.2
1.2.1
1.2: Finding Limits Graphically and Numerically
Limit of a function:
Definition of a Limit:
lim f ( x)
x
a
L
The statement above means that we can make the values of f ( x) arbitrarily close to L by taking x
to be sufficiently close to a but not equ
2.1.1
2.1: The Derivative and the Tangent Line Problem
What is the definition of a tangent line to a curve?
To answer the difficulty in writing a clear definition of a tangent line, we can define it as the
limiting position of the secant line as the secon
1.5.1
1.5: Infinite Limits
There are two types of limits involving infinity.
Limits at infinity, written in the form lim f ( x) or lim f ( x ) , are related to horizontal asymptotes
x
x
and will be covered in Section 3.5, as we learn to graph functions.
I
2.2.1
2.2: Basic Differentiation Rules and Rates of Change
Basic differentiation formulas:
1.
d
(c) 0 for any constant c.
dx
2.
d n
(x )
dx
3.
d
cf ( x)
dx
4.
d
f ( x) g ( x)
dx
d
f ( x)
dx
d
g ( x)
dx
5.
d
f ( x) g ( x)
dx
d
f ( x)
dx
d
g ( x)
dx
nx n
c
3.6.1
3.6: A Summary of Curve Sketching
Steps for Curve Sketching
1. Determine the domain of f.
2. Find the x-intercepts and y-intercept, if any.
3. Determine the end behavior of f, that is, the behavior for large values of x (limits at
4.
5.
6.
7.
8.
9.