In Exercise, recall that for varying quantities produced and sold over time period t, the revenue function is R(t) = p(t) · q(t).
Finance—Market Analysis (continuation of Exercise 1): Use the revenue function from Exercise 1 to complete the following.
(a) Verify that after multiplying and simplifying R′(t) in Exercise 1, we get the function R′(t) = 68t3 − 3060t2 − 2200t + 66,000.
(b) Graph R′, and use the Zero command on your calculator to determine the t-value so that R′(t) = 0 on the interval 0 ≤ t ≤ 7.
(c) Graph R and use the dy/dx or Draw Tangent command on your calculator to calculate the slope of the line tangent to the graph of R at the value you found in part (b).(Hint: The slope should be close to zero.)
(d) Is the revenue maximized or minimized at the t-value you found in part (b)? Explain your answer.
Exercise 1
Finance—Market Analysis: Market analysts at the Maxwell Company have estimated that the monthly sales during the first seven months of its new MePad tablet computer can be modeled by
q(t) = 30t − 0.5t2 0 ≤ t ≤ 7
where t represents the number of months since the MePad became available and q(t) represents the number of computers sold in hundreds. The analysts also determine that the price of the MePad will vary and sets the price using the model
p(t) = 2200 − 34t2 0 ≤ t ≤ 7
where t represents the number of months since the MePad became available and p(t) represents the price of the MePad in dollars.
(a) Compute q(3) and q′(3) and interpret each.
(b) Compute p(3) and p′(3) and interpret each.
(c) Write the revenue function R′(t). Do not algebraically simplify the function.
(d) Compute R(3) and R′(3) and interpret each.
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