1. A demand function is given by P = 24 – 6 ln Q
(a) Write down the equation for TR. Determine the value of Q at which TR is maximized
(b) Write down the equation for MR. Show, by differentiation, that MR decreases but
never reaches zero; that MR is concave up
(Teresa Bradley., Essential Mathematics for Economics and Business., Wiley., Fourth
Edition., page 345., number 1)
2. A consumption finction is given by the equation 𝐶 = 800 (1 − 𝑒 −0.2 𝑌 )
(a) Write down the equation for the MPC. Hence describe how consumption changes as
income increases
(b) Use differentiation to show the consumption has no maximum value
(c) Plot the consumption function. How would you describe the curvature of the
consumption function ? Use the second derivative to confirm your answer
(Teresa Bradley., Essential Mathematics for Economics and Business., Wiley., Fourth
Edition., page 346., number 2)
3. A utility function is given by the equation 𝑈 = 20 𝑥 𝑒 −0.1 𝑥 , where x is the number of
glasess of wine consumed
(a) Show that this utility function ha s a maximum value and calculate the maximum
utility
(b) Describe how marginal utility changes for glasses of wine consumed after the
maximum utility reached. Do yo consider this reasonable ? give an explanation
(Teresa Bradley., Essential Mathematics for Economics and Business., Wiley., Fourth
Edition., page 346., number 3)
4. A firm’s average revenue function is AR = 50⁄ 0.05𝑄
𝑒
(a) Determine the equations for TR and MR
(b) Find the value of Q fior which TR is maximum. Calculate the price and TR when TR is
maximized
(Teresa Bradley., Essential Mathematics for Economics and Business., Wiley., Fourth
Edition., page 346., number 4)
5. The enjoyment derived from watching a particular film is given by the equation 𝑈 = 95 −
ln(𝑡 + 5), where t is the viewing time in minutes
(a) Graph the utility function for the first two hours of viewing time. Hence describe how
enjoyment changes as the film progress
(b) Determine the equation for marginal utility and plot its graph
(Teresa Bradley., Essential Mathematics for Economics and Business., Wiley., Fourth
Edition., page 346., number 5)
6. Given 𝐴𝐶 =
100 𝑒 0.05 (𝑄−10)
𝑄+1
(a) Find the minimum AC and graph the AC function for Q=0 to Q = 25
(b) Write down the equation of TC
(Teresa Bradley., Essential Mathematics for Economics and Business., Wiley., Fourth
Edition., page 346., number 6)
7. (a) Differentiate the following,
(i) 𝑦 = 4 + 𝑥 4.5 +
2𝑥
√𝑥
1
(ii) 𝑦 = 80 𝑒 8𝑡
(a) The demand function for safari holidays is given by the equation 𝑃 = 8000 − 20𝑄,
where P is the price of the holiday and Q is the number of travelers
(i) Write down the equation for total revenue and marginal revenue
(ii) Calculate the number of travelers on a safari (Q) for which the total revenue is
maximized. Calculate the maximum revenue
(Teresa Bradley., Essential Mathematics for Economics and Business., Wiley., Fourth
Edition., page 346., number7)
8. (a) Dffrentiate the following (i) 𝑦 = 20 (1 − 𝑒 −2𝑡 ), (ii) 𝑄 = 3𝐿0.8 + 3
2
√𝐿
(b )The demand function for hand- painted T-shirt is given by the equation 𝑃 = 1600 − 4𝑄
while the average cost function is 𝐴𝐶 =
500
+
𝑄
3 (i) Write down the equations for total
revenue and marginal revenue (ii) Calculate the quantity which must sold to maximize
total revenue. Calculate the maximum revenue (iii) Confirm your answer in (ii) by
graphing the total revenue function
© (i) Write down the equation of total cost and marginal cost (ii) write down the equation
for profit (iii) Calculate the maximum profit. How many T shirt must be produced and
sold to maximize profit?
(Teresa Bradley., Essential Mathematics for Economics and Business., Wiley., Fourth
Edition., page 346., number 8)
9. (a) Differentiate the following (i) 𝑦 = ln(𝑥 2 + 5𝑥) (ii) 𝑦 = (3 + 𝑥)𝑒 𝑥 . Hence determine the
maximum and minimum points using the slope test
(b) A firm has average cost function 𝐴𝐶 =
125
𝑄2
+ 16
𝑄
− 4, (i) Calculate the value of Q for
which the average costs are minimum. Hence state the range of values for which
average costs are decreasing (ii) Write down the equation for total cost. Hence
calculate the value of Q for which total costs are minimum
(a) If a government subsidy of 3 per unit produced is given, write down the equation for
total cost. Hence calculate the value of Q for which total costs are minimum
(Teresa Bradley., Essential Mathematics for Economics and Business., Wiley., Fourth
Edition., page 347., number 9)
10. See question 2, progress exercise 4.8
(a) Diffrentiate the following
𝑆 = 200.000 (1 − 𝑒 −0.05𝑡 )
(b) The sales og magazine are expected to grow according to the equation in (a) where t
is ini weeks. Show by differentiation that sales never reach an exact maximum value.
What is the limiting sales value approached?
(Teresa Bradley., Essential Mathematics for Economics and Business., Wiley., Fourth
Edition., page 347., number 10)
11. The demand function for a monopolist is Q = 120 – 3 P
(a) Find expression for TR, MR and AR. Evaluate TR,MR and AR at Q=15. Hence explain in
words meaning of each function at Q=15
(b) Calculate the value of Q for which AR=0. Calculate the value of MR when AR = 0. Is it
sensible for a monopolist to sell this value of Q
(Teresa Bradley., Essential Mathematics for Economics and Business., Wiley., Fourth
Edition., page 280., number 1)
12. The demand function for a good is 𝑃 = 125 − 𝑄1.5
(a) Find expressions for TR, MR and AR. Is the slope of MR curve twice the slope of the AR
curve?
(b) Evaluate TR, MR and AR at Q=10 and Q = 25. Hence, explain in words the meaning of
each function at Q= 10 and Q = 25
(c) Calculate the value of Q for which MR = 0
Calculate the value of Q for which AR = 0
At what point (value of Q) does the sale of further units first start to reduce the total
revelue? Explain
(d) Graph TR, MR and AR on the same diagram
(Teresa Bradley., Essential Mathematics for Economics and Business., Wiley., Fourth
Edition., page 280., number 2)