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Business calculus final exam review
Business calculus final exam review
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Answers
1
second derivate is positive = concave down
True
False
2
\int_{ }^{ }k\cdot f\left(x\right)dx=\frac{1}{k}\int_{ }^{ }f\left(x\right)dx
True
False
3
\frac{d}{dx}\left(e^x\right)=_____
xe^{x-1}
0
1
e^x
4
P=profit, R=revenue, C=cost
True
False
5
When estimating the area under the curve, the variable Δx represents ________
how tall the rectangles are
the width of the rectangles
what type of estimate to do
the number of rectangles
6
In the context of profit, revenue, and cost, what does p represent?
Profit
the number of units
demand
Marginal revenue
7
\int_{ }^{ }x^ndx=nx^{n+1}
False
True
8
How do we find which function is on top when solving for area between curves?
test points
take the antiderivative
set the two functions equal
take the derivative
9
What is not a type of estimate we have done for areas?
midpoints
right endpoints
left endpoints
back endpoints
10
All of the derivative rules can NOT be used together
True
False
11
second derivate is negative = concave down
True
False
12
How many different variables can be in an integral?
1
2
as many as you want
2
13
What happens if you get more than 2 bounds when finding area between curves?
you have more than one integral
you throw them out until you only get two
you use the smallest and largest one
this isn't possible
14
What is the first thing you should always do first when evaluating a limit?
graph the equation
take the derivative
plug in
algebra
15
\frac{d}{dx}\left(\ln\left(x\right)\right)=_____
\frac{1}{x}
x
1
0
16
1/x = _____
-1/x2
x1/2
0
x-1
17
Select the correct order for finding where a function is concave up/concave down?
number line, set equal to zero, solve for x, take second derivative
take second derivative, set equal to zero, solve for x, number line
set equal to zero, solve for x, take second derivative, number line
set equal to zero, solve for x, number line
18
This is the equation for the chain rule for derivatives
False
True
19
Which is NOT a condition for a function being continuous at x=a?
the limit as x approaches a equals f(a)
the limit as x approaches a exists
f(a) is defined
f(a)=0
20
\frac{d}{dx}\left(e^{2x+1}\right)=e^{2x+1}_____
True
False
21
What is the name of this?
Fundamental Theorem of Calculus
Fundamental Theorem of Area
Fundamental Theorem of Antiderivatives
Fundamental Theorem of Integrals
22
How do you find the marginal revenue function?
take the antiderivative of the profit function
take the derivative of the profit function
take the derivative of the revenue function
take the derivative of the cost function
23
You can take the integral of a multivariable function.
True
False
24
\sqrt{x} = _____
0
x-1
-1/x2
x1/2
25
What is the quotient rule for derivatives?
\frac{d}{dx}\left[\frac{f}{g}\right]=\frac{gf'+fg'}{g^2}
\frac{d}{dx}\left[\frac{f}{g}\right]=\frac{gf'-fg'}{g^2}
\frac{d}{dx}\left[\frac{f}{g}\right]=\frac{gf'-fg'}{g}
\frac{d}{dx}\left[\frac{f}{g}\right]=gf'-fg'
26
What are these estimates of areas called?
integrals
area
sums
Riemann sums
27
You can take the integral of a multivariable function.
False
True
28
What tells us about increasing/decreasing of a function?
second derivative
the antiderivative
first derivative
the domain
29
You can have variables in your bounds of a double integral.
False
True
30
A vertical asymptote occurs when a limit as x goes to infinity equals a constant
True
False
31
What is different about using implicit differentiation from taking a derivative normally?
you don't use normal derivative rules
\frac{dy}{dx}does not appear
\frac{dy}{dx}appears
you can't use algebra
32
Where do relative extrema occur?
where the function changes sign
where the function is undefined
where the second derivative changes sign
where the first derivative changes sign
33
Is this equation correct?
False
True
34
\ln\left(1\right)=0
False
True
35
\int_{ }^{ }kdx=kx+c
True
False
36
When finding the area between curves, a and b represent ________
where the two functions intersect
whatever we want them to
where the two functions do not intersect
where the two functions do not exist
37
What does F represent?
the antiderivative of f
the derivative of a
the derivative of f
the derivative of b
38
How do you evaluate a double integral?
do a u-sub
Take the outside integral first
Take the inside integral first
you can't
39
What is the product rule for derivatives?
\frac{d}{dx}\left[f\cdot g\right]=\frac{f'g}{fg'}
\frac{d}{dx}\left[f\cdot g\right]=f'g-fg'
\frac{d}{dx}\left[f\cdot g\right]=f'g+fg'
\frac{d}{dx}\left[f\cdot g\right]=f'g\cdot fg'
40
Once you have your integral to find area between curves, you should NOT simplify before taking the antiderivative
False
True
41
How do we find the bounds when solving for area between curves?
take the antiderivative
test points
take the derivative
set the two functions equal
42
You can't find a higher order partial derivative.
False
True
43
\frac{d}{dx}(constant)=_____
0
the constant
1
-1/x2
44
In the context of profit, revenue, and cost, what does x represent?
Marginal revenue
demand
Profit
the number of units
45
U-substitutions are _____________ the chain rule
the constants of
the same as
not related to
the opposite of
46
How do you find the marginal cost function?
take the derivative of the profit function
take the derivative of the revenue function
take the derivative of the cost function
take the antiderivative of the profit function
47
Does the function value matter when evaluating a limit?
all of the above
no
sometimes
yes
48
How do you find the marginal profit function?
take the derivative of the cost function
take the derivative of the revenue function
take the derivative of the profit function
take the antiderivative of the profit function
49
What shape do we use the estimate the area under the curve?
triangles
squares
rectangles
circles
50
Select the correct order for finding where a function is concave up/concave down?
take second derivative, set equal to zero, solve for x, number line
set equal to zero, solve for x, number line
number line, set equal to zero, solve for x, take second derivative
set equal to zero, solve for x, take second derivative, number line
51
A horizontal asymptote occurs when a limit as x goes to a constant equals infinity
True
False
52
What is the power rule for derivatives?
0
\frac{d}{dx}\left(x^n\right)=x^{n-1}
\frac{d}{dx}\left(x^n\right)=nx^{n-1}
\frac{d}{dx}\left(x^n\right)=nx^n
53
Antiderivatives are _____________ derivatives
not related to
the opposite of
the same as
the constants of
54
Integrals give us the area under the curve
True
False
55
What tells us about the concavity of a function?
first derivative
the antiderivative
second derivative
the domain
56
The closed interval method is used to find absolute extrema.
False
True
57
Select the correct order for finding where a function is increasing/decreasing?
take derivative, set equal to zero, solve for x, number line
number line, set equal to zero, solve for x, take derivative
set equal to zero, solve for x, take derivative, number line
set equal to zero, solve for x, number line
58
What does the variable A stand for in continuously compounded interest?
dollar amount in the account
positive
prime
Principle aka initial investment
59
\int_{ }^{ }\frac{2}{x}dx=2\ln\left|x\right|+c
True
False
60
What function relates P, R, and C correctly?
R=PC
P=R-C
P=R/C
C=R-P
61
What does a limit predict?
the range of a function
the domain of a function
the x-value of a function at a certain y-value
the y-value of a function at a certain x-value
62
What is the revenue function?
p=Rx
Marginal revenue
P=R-C
R=px
63
How do we take the derivative of a multivariable function?
partial derivatives
you can't
using implicit differentiation
like normal
64
What needs to be true for a limit to exist?
The left hand limit equals the right hand limit
The left hand limit does NOT equal the right hand limit
it equals infinity
it equals zero
65
How do you find the area between curves?
\int_a^b\left[top\cdot bottom\right]dx
\int_a^b\left[\frac{top}{bottom}\right]dx
\int_a^b\left[bottom-top\right]dx
\int_a^b\left[top-bottom\right]dx
66
Δx
True
False
67
\int_{ }^{ }\frac{2}{x}dx=2\ln\left|x\right|+c
False
True
68
What does the variable P stand for in continuously compounded interest?
positive
dollar amount in the account
Principle aka initial investment
prime
69
When estimating the area under the curve, the variable n represents ________
the number of rectangles
what type of estimate to do
how tall the rectangles are
the width of the rectangles
70
\frac{d}{dx}\left(\ln\left(2x^2+1\right)\right)=\frac{4x}{2x^2+1}
True
False
71
second derivate is negative = concave down
False
True
72
What is NOT a limit evaluation technique?
plug in
multiply by the radical conjugate
factor and cancel
take the derivative
73
How many variables do you actually treat as a variable when finding a partial derivative?
2
all of them
1
0
