MATH 119 Final Exam List (Part 1)
1) Each of the function in the following table is increasing or decreasing in different way. Which of the graphs below best fits each function
Graph AGraph BGraph CGraph D
t / g(t) / h(t) / k(t) / f(t)1 / 20 / 30 / 20 / 30
2 / 22 / 26 / 30 / 22
3 / 26 / 20 / 38 / 16
4 / 32 / 12 / 44 / 12
5 / 40 / 2 / 48 / 9
Graph
2) Determine whether each of the following tables of values could correspond to a linear function or exponential function, or neither. If it is linear or exponential, find the formula for the function and then find it at t = 10.
t / g(t) / h(t) / k(t)0 / 12 / 20 / 20
1 / 10 / 19 / 22
2 / 8 / 18.05 / 24.2
3 / 6 / 17.1475 / 26.62
Formula
Estimate each at t =10
3) Given the following functions, find the graph that best represnts each function:
function / / /Best represented by Graph
Graph AGraph B Graph CGraph DGraph E
4) Suppose a town has a population of 10,000. Fill in the values of the population in the table if:
a) each year, the town’s population grows at a rate of 500 people per year.
b) each year, the town’s population grows at a rate of 5% per year.
Year / 0 / 1 / 2 / 3Population grows at a rate of 500 per yr. / 10,000
Population grows at a rate of 5% per yr. / 10,000
5) The price P of an item increased from $6,000 in 1970 to $9,000 in 1990. Let t be the number of years since 1970 (i.e. t = 0 corresponds to the year 1970).
a) Find the equation for P assuming that the increase in price has been linear.
b) Find the equation for P assuming the increase in price has been exponential.
c) Fill in the following table
t / Price P (Linear Growth ) / Price P (Exponential Growth )0 / $6,000 / $6,000
20
30
6) Give a possible formula for the following function in the form of :
7) The total cost C of producing q units of a certain item is tabulated below :
Total cost: C / 20 / 25 / 30 / 35Number of units produced: q
/ 0 / 2 / 4 / 6a) What is the fixed cost?
b) Find the linear equation which expresses the total cost C as a function of q.
c) Find the total cost for producing q = 10 units.
d) Find the linear equation which expresses q as a function of the total cost C.
e) How many units can be produced at a total cost of $40?
8) A certain hand-held calculator is being sold by the manufacturer at a price of $90 per unit. The fixed cost for production is $120,000 and each unit costs $30 to make. Let q be the number of units sold.
a)Write the following:
revenue function R(q):
cost function C(q):
profit function P(q):
b) How many units the manufacturer needs to sell to break even?
9) Using the following prices of a book, Estimate the price in the year 2000.
1990: $51991: $6
1992: $7.21993:$8.64.
10) A $ 30,000 truck has a resale value of $10,000 ten years after it was purchased.
a) Find the formula of the value of the truck as a function of time
b) Sketch a graph of the value
c) When will the value of the truck be $0?
11) Suppose that the demand and Supply function for a product is given by:
q = -p + 8 and q = 2p + 2
where p is the unit price in $ of the product.
a) Find the equilibrium point and the quantity of the product
b) graph the two functions, lable the demand and supply function
and show the shortage and surplus area
12) A movie theater owner found that when the price for a ticket was $25, the average number of customers per night was 500. When the price was reduced to $20, the average number of customers went up to 650.
a) Find the formula for the demand function, assuming that it is linear
b) Find the number of customers when the price is $5
13) One of the following tables represents supply curve and the other represents demand curve:
q / 10 / 22 / 35 / 45 / q / 40 / 32 / 25 / 15p / 5 / 10 / 15 / 20 / p / 5 / 10 / 15 / 20
a) At a price of $10, how many items would the consumers purchase? ______
b) At a price of $10, how many items would the manufactures supply? ______
c) Will the market push the prices higher or lower than $10? Why?
14a) Solve for t for each of the following equations (you must show your work):
a)b)
c) ln(t – 1) = 2d) ln(3t + 1) + ln (3t – 1) = 0
e) f)
g) ln t =2h) ln(3t - 1) - ln (2t + 1) = 0
14b) i) Convert the function to the form
j) Convert the function to the form
15) You open an IRA account with an initial deposit of $8,000 which will accumulate taxfree at 4 % per year, compounded continuously.
a) What is the effective annual yield?
b) How much (to the nearest penny) will you have in your account after 10 years?
c) How long does it take your initial investment to double?
16) If 200 people have a cellular telephones in a company of 1000 employees. If the number of cellular phones was growing at 10% a year and the number of employees at 2% per year. How long will it take to have one cellular phone per employee? (assume continuous growth)
17) A fishery stocks a pond with 2000 young trout. The number of the original trout still alive after t years is given by:
a) How many trout left after 6 months
b) At what time will there be 200 of the original trout left?
18) Find the present value of $100,000, due thirty years from now, if interest is compounded continuously at rate of 5 % per year.
19)Following the birth of a child, a parent wants to make an initial investment that will grow to $100,000 by the child’s eighteenth birthday to take care of the child’s college education expenses. Assuming that the annual rate of return on the investment is 7%, compounded continuously, what should be the initial investment? Show your work.
20)It is determined that the value of a certain computer declines exponentially. A computer purchased 2 years ago for $5,000 is worth only $2,500 today. What will the value of the computer be 2 years from now?
21)At what interest rate, when compounded continuously, will an investment double in 5 years?
22) How long does it take amount to double at 8.5% compounded:
a) annuallyb) continuously
23) The half-life of a certain radioactive substance is 10 days. If there are 8 grams initially:
a) Find the rateb) when will the substance be reduced to 2 grams?
24) If the quantity of a certain radioactive substance is decreases by 5% in 10 hours, find the half-life.
25) The population of a certain town is declining exponentially due to immigration. If only 80% of the original population are still in town after 10 years:
a) Find the decline rate.
b) How long will it take for the population to be half what it was?
26) Use the graph on the right to sketch following:
a) the line segment corresponding to ;
label that line segment as line A;
b)the line whose slope is given by;
label that line as line B;
c)the line whose slope is given by ;
label that line as line C;
27) The population of a town in millions is given by: P = 1.2(1.01)t where t is the number of years since the start of 1998 (i.e. t = 0 corresponds the year 1998). Find:
a) The population in 2000
b) The average rate of growth between 1998 and 2000:
c) How fast the population is growing at the start of 1998? (Hint: Estimate the instantaneous rate of change of P at t = 0 using h = 0.01)
28) The distance s traveled by an object as a function of time t is given in the following table:
t (sec) / 0 / 1 / 2 / 3 / 4 / 5s (feet) / 0 / 2 / 5 / 9 / 15 / 27
a) Find the average velocity of the object between t = 1 and 4.
b) Estimate the velocity of the object at t = 3. (you can use one interval only)
29) Draw a possible graph for the following functions (just show the shape of the graph):
a) s(t) = mt - 4 where m > 0b) s(t) = mt + 4 where m < 0
c) s(t) = 5(a)t where a > 1d) s(t) = 3(a)t where a < 1
30) If f(x) = x2 - 2x, find f ' (2) .(use only h = 0.001 and show all steps)
31) Draw a possible graph of given the following information about its derivative:
a) for x < 1 and x > 3for1 < x < 3
at x = 1 and x = 3 / b)f ‘(x) < 0on x < 2 and 4 < x < 6
f ‘(x) = 0at x = 2 and x = 4 and x = 6
f ‘(x) > 0on 2 < x < 4 and x > 6
32) a) Using the following graph, estimate the intervals or points where:
andand
b)
33) Suppose that f(t) is a function, that f(25) = 4 and that f ’ (25) = -0.2. Use this information to estimate:
a) f(26)b) f(28)
34) A company’s revenue R is a function of advertising expenditure, a . Suppose R = f(a)
a) What does f ’(100) = 0.8 mean?
b) If f ‘(100) = 0.8, should the company spend more or less in advertising? Why?
35) Draw a possible graph of a function whose:
a) Second derivative is everywhere negative but first derivative is everywhere positive
b) Second derivative is everywhere positive but first derivative is everywhere negative
c) Second derivative is everywhere negative and first derivative is everywhere negative
d) Second derivative is everywhere positive and first derivative is everywhere positive
36) The cost and revenue functions for a company is shown the following future, Use the Marginal revenue and Marginal cost to answer the following questions:
a) Should the company produce the 100th unit? why?
b) Should the company produce the 300th unit? why?
c) The maximum profit is at ______why?
37) The graph above shows the Cost and Revenue functions associated with a certain product. Give the value(s) of q for which
What is the value of q that will maximize profit?
After producing 30 units, should the manufacturer produce more? Why?
After producing 60 units, should the manufacturer produce more? Why?
38) The population of a town in millions is given by: P = 1.5(1.01)t where t is the number of years since the start of 1997 (i.e. t = 0 corresponds the year 1997). Find:
a) The average rate of growth between 1997 and 2001
b) How fast the population is growing at the start of 2001? (Hint: Estimate the instantaneous rate of change of P at t = 4 using h = 0.01)
39) Sketch the graph of the first and second derivatives of the functions given below. Be sure that your sketches are consistent with the important features of the original functions.
a)b)
c)d)
40) Referring to the graph of a derivateof a function for given below, indicate the points or intervals where the following conditions can hold.
The Original function is Increasing in:The Original function is Decreasing in:
The Original function has Inflection Points at:
The Original function has Horizontal Tangent line at:
41) Values of a function are given in the table to the right.
a) Estimate from left and from right then average them
Left sum:Right sum:
b) For your estimate in part (a), what is n? what is
42) Estimate the value of the definite integral by using n = 4 and computing:
xa) The left hand sum
b) The right hand sum
43) Estimate the value of the definite integral by using n = 8 and computing:
xa) The left hand sum
b) The right hand sum
44) Graph the region whose area is given by the definite integral . Estimate the area using the left sum with n = 4.
x45) Graph the region whose area is given by the definite integral . Estimate the area using the left sum with n = 4.
x46) Suppose that r =10( 1.5t ) where r is the rate in which the world’s oil is being consumed. Using 3 subdivisions, find an approximate value for the total quantity of oil used between t = 0 in 1995 and in 1998 Using a left hand-sum only.
47) Suppose that the velocity of an object is given by , where t is in seconds. Estimate the distance traveled by the object during the first 5 seconds (that is, between t = 0 and t = 5) using n = 5
a) the left sum:
b) The right sum
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