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Applied Calculus Exam 2, Summer '02: Solutions and Graphs, Exams of Calculus

Aliah UniversityCalculus

Prof. Brick's applied calculus exam 2 solutions for summer '02, including inflection points, demand elasticity, drug concentration, riemann sum, toy rocket velocity, critical points, ticket pricing, cost analysis, and logistic population growth. Each problem is solved algebraically and graphically.

Typology: Exams

2012/2013

Uploaded on 03/31/2013

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lakshmicunt 🇮🇳

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Prof. S. Brick Math 120
Summer ’02 Applied Calculus; Exam 2 section 11
Note: Be sure to read each problem completely. Show all your work. Justify all your
answers. Do the problems in order in your bluebook.
1. Algebraically find the two inflection points of y=e−x2. Sketch a graph showing them.
2. Suppose demand is given by q= 500 −10p. Is demand elastic or inelastic at p= $20 ?
How should you change the price if you want to increase revenue ?
3. Suppose the surge function y= 10te−0.25tmodels the concentration (in mg/l) of the drug
mathsmartatol. Suppose the drug is effective provided the concentration is at least 7 mg/l.
Use your graphing calculator to find the approximated values of the time coordinates that
bound the region of drug effectiveness. Sketch the graph.
4. Using the right-hand rule with 4 evenly-spaced rectangles set up (but do not compute)
the Riemann sum for Rπ
0sin(x)dx. Sketch a graph showing the rectangles. Also, using the
Fundamental theorem find the exact value of the integral.
5. Suppose a toy rocket is launched into the air from the ground. Its flight is a path 100 feet
straight up and then down and lasts, from start to finish, a total of 5 seconds. If v(t) is its
velocity where upwards is postive and downwards is negative, explain why R5
0v(t)dt = 0.
6. Find and classify the critical points of f(x) given the graph of f0(x):
7. You sell tickets for an exciting math lecture (wow!!). If you charge $20 you sell 1000
tickets. Each dollar increase in price results in 100 fewer sales. What price results in
maximum revenue ? (Use techniques from class. Do not just make up a table of values.)
8. Suppose cost in thousands of dollars is given by C(q) = q3
−12q2+ 60qwhere qis
quantity in thousands of items. Determine whether or not at q= 2 production be increased
or decreased in order to lower average cost. Draw a graph of C(q) illustrating the key fact
about marginal cost versus the average cost at q= 2.
9. Suppose the ecosystem of an island has carrying capacity of 700 rabbits. If 350 rabbits
are introduced, sketch a possible logistic model for the population as a function of time.
10. Find the derivative of f(x) = 2x5+ 2π5
x·ln(x).

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Prof. S. Brick Math 120

Summer ’02 Applied Calculus; Exam 2 section 11

Note: Be sure to read each problem completely. Show all your work. Justify all your answers. Do the problems in order in your bluebook.

  1. Algebraically find the two inflection points of y = e−x 2 . Sketch a graph showing them.
  2. Suppose demand is given by q = 500 − 10 p. Is demand elastic or inelastic at p = $20? How should you change the price if you want to increase revenue?
  3. Suppose the surge function y = 10te−^0.^25 t^ models the concentration (in mg/l) of the drug mathsmartatol. Suppose the drug is effective provided the concentration is at least 7 mg/l. Use your graphing calculator to find the approximated values of the time coordinates that bound the region of drug effectiveness. Sketch the graph.
  4. Using the right-hand rule with 4 evenly-spaced rectangles set up (but do not compute) the Riemann sum for

∫ (^) π 0 sin(x)^ dx. Sketch a graph showing the rectangles. Also, using the Fundamental theorem find the exact value of the integral.

  1. Suppose a toy rocket is launched into the air from the ground. Its flight is a path 100 feet straight up and then down and lasts, from start to finish, a total of 5 seconds. If v(t) is its velocity where upwards is postive and downwards is negative, explain why

∫ (^5) 0 v(t)^ dt^ = 0.

  1. Find and classify the critical points of f (x) given the graph of f ′(x):
  2. You sell tickets for an exciting math lecture (wow!!). If you charge $20 you sell 1000 tickets. Each dollar increase in price results in 100 fewer sales. What price results in maximum revenue? (Use techniques from class. Do not just make up a table of values.)
  3. Suppose cost in thousands of dollars is given by C(q) = q^3 − 12 q^2 + 60q where q is quantity in thousands of items. Determine whether or not at q = 2 production be increased or decreased in order to lower average cost. Draw a graph of C(q) illustrating the key fact about marginal cost versus the average cost at q = 2.
  4. Suppose the ecosystem of an island has carrying capacity of 700 rabbits. If 350 rabbits are introduced, sketch a possible logistic model for the population as a function of time.
  5. Find the derivative of f (x) =

2 x^5 + 2π^5 x · ln(x)