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MAT321 Final Exam Solutions May 1999 - Prof. William Thistleton, Exams of Calculus

State University of New York Polytechnic - Utica-RomeCalculus
Prof. William Thistleton

Solutions to various mathematical problems from a university-level final exam, including finding the center, foci, and vertices of an ellipse, evaluating derivatives, converting between polar and rectangular coordinates, analyzing critical numbers and extrema of a function, and finding the equation of a tangent line.

Typology: Exams

Pre 2010

Uploaded on 08/09/2009

koofers-user-04d
koofers-user-04d 🇺🇸

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MAT321 Final 5 May, 1999
Prof. Thistleton
1. Consider the ellipse described by the equation
4x2−16x+ 16y2−32y= 32.
(a) Where is the center of this ellipse?
(b) Where are the foci of this ellipse?
(c) Where are the vertices of this ellipse?
(d) Make a quick sketch of the ellipse on the axes below.
1
pf3
pf4
pf5

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MAT321Prof. Thistleton1. Consider the ellipse described by the equation Final 5 May, 1999 4 x^2 − 16 x + 16y^2 − 32 y = 32. (b) Where are the foci of this ellipse?^ (a) Where is the center of this ellipse? (d) Make a quick sketch of the ellipse on the axes below.(c) Where are the vertices of this ellipse?

  1. Evaluate the following derivatives (show(a) f (x) = x√x + 1 all work and simplify your results): (b) y(x) = x x+1− 2 (c) g(x) = sin(x^2 + x + 1) (d) f (x) = sec^2 (x) (e) h(x) = x^2 (x^2 + 4)^2
  1. Consider the function(a) Find the critical numbers of f (x). f (x) = (^) x (^2 1) + 1

(b) Where is f (x) increasing and where is it decreasing? (c) Where is f (x) concave up and where is concave down? (d) Find all extrema on the interval − 2 ≤ x ≤ 3.

  1. Find the equation of the line tangent to the graph of the functionpoint (x 0 , y 0 ) = (1, 1 /2). f (x) = x/(x^2 + 1) at the

Using differentials, approximate √80.