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Global Extrema - Calculus - Exam, Exams of Calculus

Punjab Engineering CollegeCalculus

This is the Exam of Calculus which includes Justification, Irrelevant, Irrelevant Statements, Inverse Trigonometric Functions, Interval, Initial Value etc. Key important points are: Global Extrema, Coordinates, Local Extrema, Classify, Local Maximum, Local Minimum, Global Minimum, Global Maximum, Inflection Points., Cylindrical Container

Typology: Exams

2012/2013

Uploaded on 03/06/2013

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Math 105: Review for Final Exam, Part II
1. Consider the function f(x) = x6
−2x3on the interval [−2,2].
(a) Find the x- and y-coordinates of any and all local extrema and classify each as a local maximum
or local minimum.
(b) Find the x- and y-coordinates of any and all global extrema and classify each as a global maximum
or global minimum.
(c) Find the x-coordinate(s) of any and all inflection points.
2. Your company is mass-producing a cylindrical container. The flat portion (top and bottom) costs 3
cents per square inch and the curved (lateral) portion costs 5 cents per square inch. If your budget is
$9.00 per container, what dimensions will give the largest volume? [Students in the 1:10 section may
omit this problem.]
area of circle = πr 2lateral area of cylinder = 2πr h volume of cylinder = π r2h
pf3
pf4

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Math 105: Review for Final Exam, Part II

  1. Consider the function f(x) = x^6 − 2 x^3 on the interval [− 2 , 2].

(a) Find the x- and y-coordinates of any and all local extrema and classify each as a local maximum or local minimum.

(b) Find the x- and y-coordinates of any and all global extrema and classify each as a global maximum or global minimum.

(c) Find the x-coordinate(s) of any and all inflection points.

  1. Your company is mass-producing a cylindrical container. The flat portion (top and bottom) costs 3 cents per square inch and the curved (lateral) portion costs 5 cents per square inch. If your budget is $9.00 per container, what dimensions will give the largest volume? [Students in the 1:10 section may omit this problem.] area of circle = πr^2 lateral area of cylinder = 2πrh volume of cylinder = πr^2 h
  1. You are standing on a pier, 6 feet above the deck of a boat. Attached to the boat is a line, which you are pulling in at a rate of 3 feet per second. When there are 10 feet of line between your hand and the boat, at what rate is the boat moving across the water?
  2. Use the Intermediate Value Theorem to show that f(x) = x^3 − 2 x − 1 has a root on [1, 2].
  3. What (if anything) does the Extreme Value Theorem say about f(x) = x^2 on each of the following intervals?

(a) [1, 4]

(b) (1, 4)

  1. Find the value of the constant c that the Mean Value Theorem specifies for f(x) = x^3 + x on [0, 3].
  1. (a) Use sigma notation to express L 10 and M 10 as approximations to

20

ln x dx.

(b) Draw a sketch that represents the sum M 4.

  1. Find the following.

(a) all antiderivatives of 1 + 2x + x^3 + 4

x +

x^5

(b)

1

x

dx

(c)

− 2

4 − x^2 dx

(d)

d dx

∫ (^) x

1

sin

t dt