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Fall '04 MAT 250 Final Exam: Calculus Problems, Exams of Analytical Geometry and Calculus

Murray State UniversityAnalytical Geometry and Calculus

The final exam questions for a calculus course (mat 250) from fall '04. The exam covers topics such as differentiation, integration, limits, and optimization. Students are required to show their work and use appropriate methods like l'hospital's rule and substitution.

Typology: Exams

Pre 2010

Uploaded on 08/16/2009

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Fall ’04/MAT 250/Final Exam Name: Show all your work.
Differentiate and simplify where appropriate:
1. (5pts) d
dx
x+ 3
x2
−4=
2. (4pts) d
dx ln(sin x) =
3. (4pts) Find the limit algebraically. Do not use L’Hospital’s rule.
a) lim
x→4
x2
−2x−8
x−4=
4. (4pts) Use L’Hospital’s rule to find the limit:
lim
x→0
ex
−1−x
x2=
pf3
pf4
pf5

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Download Fall '04 MAT 250 Final Exam: Calculus Problems and more Exams Analytical Geometry and Calculus in PDF only on Docsity!

Fall ’04/MAT 250/Final Exam Name: Show all your work.

Differentiate and simplify where appropriate:

  1. (5pts)

d dx

x + 3 x^2 − 4

  1. (4pts)

d dx

ln(sin x) =

  1. (4pts) Find the limit algebraically. Do not use L’Hospital’s rule.

a) lim x→ 4

x^2 − 2 x − 8 x − 4

  1. (4pts) Use L’Hospital’s rule to find the limit:

lim x→ 0

ex^ − 1 − x x^2

  1. (8pts) Let f (x) = x^2 ex. a) Find the intervals of increase/decrease and where f has a local maximum and minimum. b) Find the intervals where f is concave up or down. c) Use your calculator and the results of a) and b) to accurately sketch the graph of f.
  2. (3pts) Find D^65 sin 3x.
  1. (7pts) Use four rectangles to estimate the area under the curve y = 9 − x^2 from x = 0 to x = 2. Choose sample points in two ways (draw a picture big and beautiful) so that you a) Overestimate the area. b) Underestimate the area.
  2. (4pts) The graph of f is given. Sketch the graph of f ′.

a b

  1. (7pts) Consider the equation ex^ − x − 2 = 0. a) Use the Intermediate Value Theorem to show that this equation has a solution in the interval [0,2]. b) Use your calculator to find an interval of length 0.01 that contains this root. Explain why the IVT will guarantee there is root in the interval that you found.
  2. (4pts) Oil is leaking from a tank at rate 3 + 2t liters per minute through a hole that is increasing in size. How much oil leaks out from t = 4 minutes to t = 7 minutes?