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Math 105: Review for Final Exam, Part II - Calculus Exercises and Problems, Exams of Calculus

Cochin University of Science and TechnologyCalculus

This is the Exam of Calculus which includes Negative, Natural Domain, Moving Across, Most General, Method, Maximum Score etc. Key important points are: Local Maximum, Coordinates, Local Extrema, Classify, Local Minimum, Global Extrema, Global Maximum, Global Minimum, Inflection Points, Deck

Typology: Exams

2012/2013

Uploaded on 03/06/2013

jugnu
jugnu 🇮🇳

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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. 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?
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. 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?
  1. Use the Intermediate Value Theorem to show that f(x) = x^3 − 2 x − 1 has a root on [1, 2].
  2. 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].
  2. Water is leaking out of a tank at a decreasing rate r(t) as shown in the table below.

time (min) 0 2 4 6 8 rate (gal/min) 15 11 8 4 3

(a) Find an overestimate and underestimate for the total amount that leaked out during these 8 minutes.

(b) Interpret the expression

2

r(t) dt in terms of the situation described above.

  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