Calculus For Business
Final Exam
The cost of owning a
home includes both fixed costs and variable utility costs. Assume that it costs
$4,865.00 per month
for mortgage and insurance payments, and it costs an
average of $3.93 per unit for natural gas, electricity, and water usage.
Determine a linear equation that computes the annual cost of owning this home
if x utility units are used.
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Write a cost function
for the problem. Assume that the relationship is linear.
Fixed cost, $43; 10
items cost $5,460 to produce.
Solve:
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A toilet manufacturer
has decided to come out with a new and improved toilet.
The fixed cost for the production of this new toilet line is $16,600 and
the variable costs are $67 per toilet.
The company expects to
sell the toilets for $156.
Formulate a function
P(x) for the total profit from the production and sale of x toilets.
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Complete the table and
use the result to find the indicated limit
If f(x) = x2
– 5, find lim (fx)
x-->0
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Use the properties of the limits to help decide
whether the limit exists. If it exists, find the value.
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Use the properties of
the limits to help decide whether the limit exists. If it exists, find the
value.
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The total cost to
produce x handcrafted wagons is C(x) = 110 + 8x – x2 + 7x3.
Find the marginal cost when x = 5.
Explanation:
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Use the quotient rule to find the derivative.
f(x) = 
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The total profit from selling x units
of cookbooks is P(x) = (4x - 9) (8x - 6).
Find the marginal average profit
function, 
'(x).
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Find the derivative
√4x + 2
Find the derivative.
√4x + 2
(4x + 2)1/2 – 1 x 4x + 2
Reduce
= 
The sales in thousands
of a new type of product are given by S(t) = 240 - 40e-0.5t,
where t represents time
in years. Find the rate of change of sales at the time when t = 2.
(Round to
the nearest tenth).
The value
of e is 2.718281828459045
Find the derivative
and plug in the value for t which = 2
240 - 40e-0.5t
240 x 0 =
0
-40 e
(-.5) .5 – 1 = 20e-.5
20(2.718281828459045)
-.5(2)
20(2.718281828459045)-1
= 7.357588823
Round to
the nearest tenth = 7.4
The ph scale
is used by chemists to measure the acidity of a solution it is a base 10
logarithmic scale.
The pH, P, of a solution and its
hydronium ion concentration in moles per liter, H, are related as H = 10-P.
Find the formula for the rate of
change 
Solution
ln h = ln (10-p)
ln h = (-p) ln (10)
(-p ln (10))
-ln(10) 
= H[-ln (10)]
= -10-p ln (10)
Find the derivative of the function.
y = log(4x)
y1 = log(4x)
= 
Simplify
Assume that the total revenue received from the sale of x items is
given by R(x) = 37 ln(3x+1),
while the total cost to produce x items is
C(x)=x/3.
Find the approximate number of items that should be manufactured so
that the profit,
R(x) - C(x) is a maximum.
P(x) = R(x) – C(x)
Write the equation 37
ln (3x + 1) - 
Now we find the derivative, set P(x) to zero, and solve for x.
Find
the derivative of 37 ln (3x + 1) - 
set P(x) to zero, and solve for x
- 
= o
111 = 
(3 x 111 = 333) and (3
/ 3 = 0)
333 = 3x + 1 Subtract 1 from both sides = 332 = 3x
Divide both sides by 3 = x = 332 / 3
x = 110.66666667
Round to 111
The annual revenue and
cost functions for a manufacturer of grandfather clocks are approximately
R(x) = 520x – 0.03x2
and C(x) = 160x + 100,000
where x denotes the
number of clocks made. What is the annual maximum profit?
Same
formula as above
P(x) =
R(x) – C(x)
520x –
0.03x2 - 160x - 100,000 (note that we subtract both numbers with
C(x)).
Now we
simplify and put it in proper order.
-.03x2
+ (520x – 160x) - 100,000 = -.03x2
+ 360x - 100,000
Now we
find the derivative and set P(x) to equal zero. = -0.06x +
360 = 0
360 =
0.06x
x = 360 /
0.06 = x = 6,000
Now we
plug 6,000 in the original formula P(x) = R(x) – C(x) which is
520x –
0.03x2 - 160x - 100,000
520(6,000)
– 0.03(6,000)2 – 160(6,000) - 100,000
3,120,000 -
1,080,000 - 960,000 - 100,000
3,120,000
- 1,080,000 = 2,040,000 – 960,000 = 1,080,000 – 100,000 =
980,000
Find the number of
units(x) that produces the maximum profit(P), if C(x) = 85 + 24x and p = 40 -
2x
P(x) =
R(x) – C(x)
First, we
need to factor P(x)
40 – 2x(x)
= 40x – 2x2
40x – 2x2
– 85 – 24x (note we subtract both numbers in C(x))
Simplify
and put back in proper order.
-2x2
+ (40x – 24x) – 85 = -2x2
+ 16x – 85
Now we
find the derivative of -2x2 + 15x – 85 = 4x + 16
Now we
set to equal zero and solve for x. = 4x + 16 =
0
16 = 4x divide both sides by 4 = x = 16 /
4
x = 4
Now we
plug 4 into the derivative 4x + 16.
4(4) + 16
= 32
A rectangular field is
to be enclosed on four sides with a fence.
Fencing costs $5 per foot for two
opposite sides, and $6 per foot for the other two sides.
Find the dimensions of
the field of area 620 ft2 that would be the cheapest to enclose.
Formula required: 5(2x) + 6 (
)
Now we simplify 10x + 6 (
) = 10x + (
)
Next we set equal to 0 = 10 - (
) = 0 and simplify - = -10
We simlify further: -10x2 =
-7440
Next we divide both sides by -10 = x2 = 744
Next we need to square 744 to solve for x: x = √744
x = 27.2763639 (round to 2 decimal places) x = 27.3
Since the dimensions of the field is equal to 620, we need to divide that by 22.3
620 / 22.3
620 / 27.3 =
22.73560689
x = 22.3, y = 27.3
Given the demand
function q = 401 - 4p calculate the elasticity of the demand when p = 53.
The
formula for this problem is:
Now
plug in the value of P (53)
212
/ 189 = 1.121693122
The
elasticity of the demand = 1.12
Given the revenue and
cost functions R(x) = 28x - 0.6x2 and C(x) = 6x + 9, where x is the
daily production, find the rate of change of profit with respect to time when
10 units are produced and the rate of change of productions is 7 units per day
28x –
0.6x2 – 6x – 9
First, we
simplify to 22x -
0.6x2 – 9
Derivative
-1.2x +
22
-1.2 x 10
+ 22 x 7
154 – 12
= 142
Evaluate the definite
integral.
∫ (5x2
− 8x + 6) dx
Apply
linearity: = 5 ∫ x2
dx – 8 ∫ x dx + 6 ∫ 1 dx
Now
solving: ∫ x2
dx Apply
power rule: ∫ xn dx = 
= 
Now
solving: ∫ x dx Apply
power rule with n = 1 =
Now
solving: ∫ 1 dx Apply
constant rule: = x
Plug in
solved integrals: 5 ∫ x2
dx – 8 ∫ x dx + 6∫ 1 dx = 
– 4x2
+ 6x
Evaluate the definite
integral.
37791 /
24010000 = 0.0015739692 = round
to 0.0016
Find the integral.
x2 (3x + x-3)
dx
Find the integral:
-7t^2 dt
∫e
Simplify 
du
Since −1 is constant
with respect to u, move −1 out of the integral.
-∫ 
du
Since 
is constant with respect to u,
move out of the integral.
∫ eu du)
The integral of eu with respect to u is eu.
− (eu + C)
Simplify. − eu + C
Replace all
occurrences of u with −7t2 − e-7t2 + C
The number of books in
a library increases at a rate according to the function B’(t) = 171e0.03t,
where t is measured in years after the library opens.
How many more books will the library have 2 year(s) after opening?
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We need to evaluate
the integral
 e0.03t
171 is constant with
respect to t,
we move 171 out of the
integral.
 e0.03t
u = 0.03t, then du =
0.03t or  du = dt
Now we rewrite the
equation.
Combine 
Next, we move  out of the integral
|
Factor and combine. 
(the integral of eu is eu
so we move that out)
e = 2.71828, and u = .006
so now we set up the equation to
factor
 (2.718280.06) -
171 / 0.03 = 5,700
5,700 x 2.718280.06 - 
6,052.468071
– 5,700 = 352.468071
Round
to 352
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The price per share of
a stock can be approximated by the function S(t) = t(30
– 30t) + 25,
where t is time (in years) since the
stock was purchased.
Find the average price of the stock over the first 8
years.
(Round the answer to two decimal places if necessary).
This is the
integral we need to solve!
You can
solve this one yourself!
This
calculator will solve it for you.
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