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sample-test1-fall-2008

# sample-test1-fall-2008 - for a function to be continuous at...

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Calculus 221 - First Test September 19, 2008 NOTE: THIS IS ONE OF SEVEN DIFFERENT EXAMS GIVEN; IT IS LIKE, BUT PERHAPS NOT THE SAME AS, THE ONE YOU TOOK. No calculators allowed. Justify your answers. 1 (a) What is the slope of the line 4 x + 5 y - 10 = 0? (b) Find the equation of the line perpendicular to the line y = 2 x - 100 which passes through the point (1 , - 6). (c) Find the center and radius of the circle x 2 + y 2 + 2 y - 3 = 0 Answers: (a) - 4 5 . (b) y = - x 2 - 11 2 . (c) center (0 , 1), radius 2. 2 Find the following limits: (a) lim h 0 4( x + h ) - 4 x h . Here you are to evaluate the limit showing details of how it’s done. (b) lim x 4 + x x 2 - 16 (c) lim x 0 x +6 - 6 x (d) Write down the function f ( x ) such that f ( x ) is the limit in Part (a). What is the domain of that function? Answers: (a) x - 1 2 (b) + (c) 1 2 6 (d) The function is f ( x ) = 4 x i.e. f ( x ) = 2 x 1 2 ; its domain is the set of all numbers 0. 3 Consider the function f ( x ) = x 3 x 2 1 5 2 < x < 3 x 3 x 3 (a) Draw the graph of this function. (b) At what values of x is it NOT continuous? 1

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2 (c) Explain your answer to (b), showing that you know what it means
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Unformatted text preview: for a function to be continuous at a point x = a . Answers: (a) The picture is the graph of y = x 3 with the part over [2 , 3] removed and replaced by a horizontal line at height 1 5 . (b) It is not continuous at x = 2 and at x = 3, (c) because the left and right limits are diﬀerent at those two places. 4 In this question derivatives may be evaluated by any (correct) method. (a) If f ( x ) = x 2-9 x 2-7 x +12 , ﬁnd f ( x ) and f (2). (b) If y = (cos x )(2 + 3 x ), ﬁnd dy dx . (c) If g ( t ) = 2 t 1100 , ﬁnd g (1). (d) If h ( u ) = cos u u , ﬁnd h ( u ). Answers: (a) f ( x ) =-7 ( x-4) 2 ; f (2) =-7 4 (b) dy dx = (-sin x )(2 + 3 x ) + 3cos x (c) g ( t ) = 2200 t 1099 (d) h ( u ) = (-u sin u )-(cos u ) u 2 5 Find the equation of the line which is tangent to the curve y = x 1+ x 2 at the point (3 , 3 10 ). Answer: y =-2 25 x + 27 50...
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