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University of Calgary - Faculty of Science

Department of Mathematics and Statistics

AMAT 217-L01 - L06 - Fall 2010 - Final Examination

Closed Book , No Calculators.

01.Let g ( x) =ax 2 + b x 2

8 x 3 x > 2

Find the values of the constant real numbers a and b such that the function g is differentiable at x = 2.

02. Find an equation of the line tangent to the curve y = x3 x 1 which passes through the point (0 , 3).

03. Determine the slope of the line tangent to the curve sin ( y 1) = x3 + xy2 2 at the point (1, 1 ) on the curve.

04.Evaluate 24 x4 + x2 + 3 x2 dx

05. Assume the function f ( x) = x2 2 x + 3 , x 1 has an inverse. Find a formula for y = f 1 ( x)

06.Let g be an inverse function of f . Using the table of values below, find g ( 12

).

x f ( x) f ( x)

2 4 5

9 12

7

07.If y = cos(e x) + e cos ( x) , find dy

dx

.

08. Find the derivative of y = [cosh ( x)] x.

09. The half life of a radioactive substance is 60 years. In how many years will three fourth of the original

amount have disintegrated?

10.If sinh ( x) = 43

, find the exact value of cosh ( x).

11.The voltage V in volts , the resistance R in ohms , and the current I in amperes are related by the

formula V = I R . The current in an electrical system is decreasing at the rate of 3 amperes / s while

the voltage remains constant at 9 volts .

At what rate is the resistance changing at the instant the current is 6 amperes ?

12. Newtons method with initial approximation x0 is used to approximate a real root of the equation

x4 2 = 0. Determine the value of Newtons method iteration x1 .

13.Evaluate x 2lim 2 x

3 3 x2 12 x + 24 x3 x2 8 x + 2

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14.Find x 2+lim (5 x2 )

1 x2

15.Let f ( x) = 3 x4 4 x3 + 5 . Find the x and y coordinates of local maxima & local minima of the function f .

16.Let f ( x) = 18( x 2) 2/3 + 9( x 2) 1/3 .The first and second order derivatives of f are respectively given by

f

( x) = 3(4 x 9)

( x 2) 4/3 , f

( x) = 4(3 x)( x 2) 7/3 . Determine the x and y coordinates of points of inflection of f .

17. Find the equations of the vertical asymptotes of

f ( x) = x2 + x + 2 2 x4 x2

.

18. Find the equations of the horizontal asymptotes of the function

f ( x) =

12 x2 x2 + 2

x < 4

5 + 6 e x

1 + 2 e x x 4

19. Find the the linearization L( x) of f ( x) = tan 1( x) about x = 1.

20.Let L( x) be the linearization of the function f about x = a. Assume that f is concave up

on an open interval containing a. Which of the following statements is true :

(I) f ( x) > L( x) for all x near a. (II) f (a ) > L(a )

(III) f ( x) < L( x) for all x near a. (IV) f (a ) < L(a )

21.Evaluate 6 x(6 x 1) 2 dx .

22.Find 1e x + e x dx

23.Determine x 0lim

02 x

(e t 2

1) dt

x3

24. Find the area of the region enclosed by the parabola y = 3 x2 x and the straight line y = 5 x.

25. Calculate the volume of the solid generated by revolving the region between the curves

y = x2 , y = 3 x , 0 x 1 , about the y axis.

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