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# c1ex5 - MAC2311 Final 1 Evaluate 1 0 y(y 2 1)5 dy 2 Find...

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MAC2311 Final 1. Evaluate 1 0 y ( y 2 + 1) 5 dy . 2. Find the absolute max and min of f ( x ) = ( x 2 + 2 x ) 3 on the interval [ - 3 , 3]. 1

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3. Evaluate the limit: lim x 0 + x 2 ln x . 4. Find the shortest distance between the line y = 3 x + 2 and the point (1,1). 2
5. Find the derivative of y = 8 e sin θ . 6. Determine y given xy 4 + x 2 y = x + 3 y . 3

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7. Find the equation of the tangent line to the function f ( x ) = x 2 - 1 x 2 +1 at the point (0,-1). 8. Evaluate π 0 x + 2 cos x dx . 4
9. The area of a triangle increasing at a constant rate of 10 cm 2 /sec . Find dh dt if db dt = 2 cm/sec and b = 3 cm , h = 2 cm . 10. For the function f ( x ) = x x +8 find: intervals upon which the function is increasing and de- creasing; intervals upon which the function is concave upward and downward; local max and mins; inflection points; and vertical and horizontal asymptotes. Use this information to graph the function. 5

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10. cont. 6
Solutions: 1. We begin with a substitution and choose u = y 2 + 1 so that du = 2 y dy or (1 / 2) du = y dy. Rewriting the integral we see y ( y 2 + 1) 5 dy = ( y 2 + 1) 5 ( y dy ) = u 5 (1 / 2) du = (1 / 2) u 5 du = (1 / 2)(1 / 6) u 6 = (1 / 12) u 6 = (1 / 12)( y 2 + 1) 6 . Therefore, 1 0 y ( y 2 + 1) 5 dy = (1 / 12)( y 2 + 1) 6 | 1 0 = (1 / 12)((1) 2 + 1) 6 - (1 / 12)((0) 2 + 1) 6 = (1 / 12)(64) - (1 / 12)(1) = 63 / 12 = 21 / 4 .

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c1ex5 - MAC2311 Final 1 Evaluate 1 0 y(y 2 1)5 dy 2 Find...

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