MAT 210 .edu
1. Find the domain of the following functions: (8pts.)
a. [pic]
Answer:
Since square root is not defined for negative numbers we have
Domain: {x | [pic]} or, in interval notation,
Domain: [-1,[pic])
b. [pic]
Answer:
Since division by zero is not defined we have
Domain: {x | [pic]} or, in interval notation,
Domain: [pic]
2. Given the functions [pic]and [pic], find [pic].
Give exact answers, no decimals. (6pts)
Answer:
[pic]
3. Telca opened a savings account with $2000. Her account has an annual yield of 6% compounded monthly. At the same time her friend Iftode opened an account at a credit union with $1000. His account has an annual yield of 9% compounded monthly.
a. Write an equation for the amount of money in Telca’s account at time t. (3pts)
Answer:
Using the formula for compound interest [pic] with P = 2000, r = .06 and m = 12 we have:
Amount of money in Telca’s account at time t: [pic]
b. Write an equation for the amount of money in Iftode’s account at time t. (3pts)
Answer:
Using again the formula for compound interest:
Amount of money in Iftode’s account at time t: [pic]
c. Algebraically determine how long it will take for the two accounts to be equal assuming no deposits or withdrawals are made. (7pts)
Answer:
We set the amount of money in the two accounts equal to each other and solve for t
[pic]
divide both sides by 1000
[pic]
divide both sides by [pic]:
[pic]
[pic]
take log of both sides:
[pic]
solve for t
[pic] years
4. The following data gives the price of a math book for certain years.
|Year |1970 |1972 |1974 |1976 |1978 |1980 |
|Price |$48.95 |$53.15 |$57.35 |$61.55 |$65.75 |$69.95 |
a. Based on this data, find the best fitting exponential function using exponential regression.
Let[pic]correspond to the year 1970. Round your answer to two decimal places. (6pts.)
Answer:
Using the ExpReg regression on the calculator and rounding the answer to two decimal places, we find
[pic]
b. Using the model obtained in part (a), estimate the price of a book in 1985. (4pts.)
Answer:
Let [pic], then [pic] dollars.
Note that if we use the regression model without rounding (e.g. use the “value” function in the calculator) the answer is 84.27 dollars.
c. Using the model obtained in part (a), estimate when the price will reach $90.00.
(6 pts.)
Answer:
We can answer this question algebraically or graphically
Algebraically: we need to solve the exponential equation [pic]
[pic]
[pic]
[pic]
so the price will reach $90.00 in 1985.
Graphically: we graph the function [pic] and the function [pic] and we find the intersection point. The two functions intersect at (15.30,90).
NOTE: if we use the regression model without rounding and we find the intersection with the line y = 90 the answer is (16.84,90). Thus, according to this model the price will reach $90.00 at the end of 1986.
5. A manufacturing firm has a daily cost function of [pic], where x is the number of thousands of an item produced and C is in thousands of dollars. Suppose that the number of items that can be manufactured is given by [pic], where t is measured in hours. Find the cost as a function of t. Interpret your answer as composition of functions. (6pts.)
Answer:
To find the cost as a function of t we just need to plug in [pic] in the formula for C(x).
This gives [pic].
[pic]is the composition of the function C with the function x, i.e.[pic]=[pic]
6. Consider the function [pic] whose graph is given below.
[pic]
a. (6pts ) Write out in words how we can obtain the graph of [pic] from the graph of [pic]. Specify the order of the transformations.
Answer:
The order of transformations is not important in this case.
The graph of [pic] is obtained from the graph of [pic] by shifting the graph one unit to the left, stretching vertically by a factor of two and reflecting in the x axis.
b. (6pts) Draw the following graphs. Be careful in labeling the x and y scales and all the important points on the graphs.
i) [pic]
Answer:
[pic] is obtained by shifting the graph of [pic] one unit to the left:
[pic]
ii) [pic]
Answer:
[pic] is obtained by stretching the graph in (i) vertically by two, i.e. by multiplying all the y values by 2.
[pic]
iii) [pic]
Answer:
[pic] is obtained by reflecting the graph in (ii) in the x axis.
[pic]
7. A typical small-sized plant produces fertilizer at the estimated fixed cost of $235,487. It costs $206.68 to produce each ton of fertilizer and the plant sells its fertilizer at $266.67 per ton.
a. Find the cost, revenue and profit equations. Clearly define the variables you are using. (9pts)
Answer:
Let x be the number of tons of fertilizer produced and sold by the plant.
The cost is given by the fixed cost plus the variable cost. Thus [pic]
The revenue is given by [pic].
Profit = revenue – cost, thus
[pic]
b. How many tons of fertilizer should they produce and sell to break even? (4pts)
Answer:
The plant will break even when revenue = cost, i.e. when profit is zero
Solving the equation [pic] we get [pic].
Thus the plant needs to sell approximately 3925 tons of fertilizer to break even.
c. What is the slope of the profit function? Interpret the meaning of the slope of the profit function in the context of the problem. Write complete sentences, be precise and do not forget your units. (4 pts)
Answer:
Since the profit function is given by [pic] the slope is [pic] $/ton.
This means that for every increase of one ton in the fertilizer production the profit will increase by $59.99.
8. Find two functions f and g such that [pic], [pic]and [pic]. (6pts)
[pic]
Answer:
There are two possible choices:
[pic], [pic]
or
[pic], [pic]
9. The table below shows the monthly profit of a concert hall for various ticket prices.
|Ticket price (dollars) |15 |18 |21 |24 |27 |30 |33 |
|Profit (millions of dollars) |13.2 |17.5 |22 |23.2 |23.1 |22.1 |18.3 |
a. Find the equation of the best model for the data. What is it? Justify your answer. (8pts.)
Answer:
By graphing the points the parabolic shape is apparent.
The quadratic regression gives [pic]. Thus the correlation coefficient is very close to one and it is also clear by graphing the regression that the fit is very good.
Cubic regression gives a slightly better [pic] but the improvement is not sufficient to justify the choice of a more complicated model such is the cubic one.
The quadratic regression also makes sense in the context of the problem: we expect the profit to raise until the price of the tickets is too high, at which point people will stop going to the concert hall.
The equation of the quadratic regression is:
[pic]
b. At what ticket price will the profit reach a maximum? (round to the nearest dollar). What will the profit be at that ticket price? (8pts.)
Answer:
We need to find the maximum (vertex) of the parabola in part a.
We can do it graphically by using the maximum feature on the calculator or algebraically by using the formula for the x-coordinate of the vertex [pic].
This gives: [pic]. Thus, rounding to the nearest dollar, the ticket price that will give the maximum profit is $26.00.
At this price the profit will be: [pic] millions of dollars.
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