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2005 Spring (short) - 125FinalSpring05shrt math125

2005 Spring (short) - 125FinalSpring05shrt math125 -...

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MATH 125, FINAL EXAM (common) May 2005 1. (6 points each) Calculate the following limits. a) lim x →- 3 x 2 - 9 x 2 + 2 x - 3 . b) lim x 0 1 - 1 - x 2 x 2 . c) lim x 8 - | x - 8 | x - 8 . d) lim x 2 + e 1 2 - x . 2. (6 points each) Find dy dx . a) y = x 2 sin 2 x . b) y = ( x - 1)( x - 4) ( x - 2)( x - 3) (You may use logarithmic differentiation). c) xe - y + ye - x = 3. d) y = x 3 cos t t dt . 3. (7 points each) Evaluate the following integrals: (a) ( 3 x - x 3 + 1 3 x ) dx . (b) 1 0 5 x 2 x 3 + 2 dx . (c) π 0 x cos( π + x 2 ) dx (d) e 2 x 1 + e x dx . 4. Consider the following function and its first and second derivative: f ( x ) = x 2 + 3 x + 1 x 2 + 1 f ( x ) = 3(1 - x 2 ) ( x 2 + 1) 2 f ( x ) = 6( x 3 - 3 x ) ( x 2 + 1) 3 . a) (4 points) Find the critical numbers of f . b) (8 points) Determine where f is increasing, where f is decreasing, and find the local maxima and minima of f . c) (4 points) Find the asymptotes of f . f ( x ) = x 2 + 3 x + 1 x 2 + 1 f ( x ) = 3(1 - x 2 ) ( x 2 + 1) 2 f ( x ) = 6( x 3 - 3 x ) ( x 2 + 1) 3 . 1
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2 d) (5 points) Find the inflection points of f and determine where f is concave upwards and downwards. e) (8 points) Draw a rough sketch of the graph of f . 5. (20 points) Find the dimension of the right circular cylinder of
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Unformatted text preview: greatest volume that can be inscribed in a given right circular cone of height 1 meter and radius 1 meter. 6. a) (10 points) Show that the equation ln x-x + 2 = 0 has at least two solutions. b) (10 points) Show that it has exactly two solutions. 7. (20 points) Find the linear approximation to f ( x ) = 1 (1 + x ) 4 at a = 0, and use it to approximate 1 1 . 1 4 . 8. (15 points) A cube is increasing in volume at a rate of 10 cm 3 /sec . Find the rate of change of the surface area of the cube when one edge has length 2cm. 9. Let f ( x ) = x 2 cot x x 6 = 0; x = 0 . a) (10 points) Show that f is continuous at x = 0. b) (10 points) Show that f differentiable at x = 0....
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