This week’s mastery quiz has three topics. Everyone should submit S8. If you already
have a 4/4 on M4, or a 2/2 on S6, you don’t need to submit it again.
Don’t worry if you make a minor error, but try to demonstrate your mastery of the
underlying material. Feel free to consult your notes, but please don’t discuss the actual quiz questions with other students in the course.
Remember that you are trying to demonstrate that you understand the concepts involved.
For all these problems, justify your answers and explain how you reached them. Do not just write “yes” or “no” or give a single number.
turn this quiz in class on Thursday. You may print this document out and write
on it, or you may submit your work on separate paper; in either case make sure your name and recitation section are clearly on it. If you absolutely cannot turn it in in person, you can submit it electronically but this should be a last resort.
Topics on This Quiz
Major Topic 4: Optimization
Secondary Topic 6: Curve Sketching
Secondary Topic 8: Riemann Sums
Name:
Recitation Section:
1
Name: Recitation Section:
Major Topic 4: Optimization
(a) Suppose that a company that produces and sells x units of a product makes a revenue of R(x) = 260x − 9×2/10 and has costs given by C(x) = 1000 + 100x + x2/10. What is the maximum profit that can be made ?
(b) Classify the critical points and relative extrema of h(x) = sin(x) + cos(x) on [0, 2π].
S6: Curve Sketching Sketch the graph of g(x) = 3×4 − 4×3 − 36×2 + 64 = (x + 2)2(3x − 4)(x − 4) have g′(x) = 12×3 − 12×2 − 72x = 12x(x − 3)(x + 2) and g′′(x) = 36×2 − 24x − 72 = 12(3×2 − 2x − 6).
You should discuss the domain, limits, critical points, intervals of increase and decrease, concavity, and possible points of inflection.
Secondary Topic 8: Riemann Sums Let f (x) = x2 − x be defined on the interval [−3, 0].
(a) Approximate the area under the curve of the function using three rectangles and right endpoints.
(b) Approximate the area under the curve of the function using three rectangles and left endpoints.
(c) Write a formula for Rn, the estimate using n rectangles and right endpoints, as a
summation of n terms.
(d) Use your answer in part (c) to find a closed-form formula for Rn.
(e) Use the formula in part (c) to compute the area exactly.
