{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Exam1_S2010 Solutions

We are given a function describing the rate of change

Info iconThis preview shows pages 5–6. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: We are given a function describing the rate of change of the population of pumas, so to find the function describing the actual population r ( t ), we must take the integral of R ( t ): r ( t ) = Z R ( t ) dt = 500 Z te . 08 t dt Use integration by parts. Choose u = t and dv = e . 08 t dt . Then du = dt and v = e . 08 t . 08 . Then integration by parts gives: 500 Z te . 08 t dt = 500 te . 08 t . 08- Z e . 08 t . 08 dt A simple u-substitution in the second integral gives r ( t ) = 500 te . 08 t . 08- e . 08 t (0 . 08) 2 + C We are told that the initial population of pumas is 1000. That is, r (0) = 500 (0) e . 08(0) . 08- e . 08(0) (0 . 08) 2 + C = 1000 This reduces to- 500 (0 . 08) 2 + C = 1000 so C = 1000 + 500 (0 . 08) 2 Thus the population function is given by r ( t ) = 500 te . 08 t . 08- e . 08 t (0 . 08) 2 + 1000 + 500 (0 . 08) 2 and the population in year 6 is r (6) = 500 (6) e . 08(6) . 08- e . 08(6) (0 . 08) 2 + 1000 + 500 (0 . 08) 2 5 9. Suppose a function f ( x ) has y-intercept (0 ,- 8), and the tangent line to the graph of f ( x ) at x = 3 has the equation 6 x- 3 y = 12. Compute Z 3 xf 00 ( x ) dx . First do the indefinite integral, using integration by parts. Choose u = x and dv = f 00 ( x ) dx . Then du = dx and v = f ( x ). Then integration by parts gives Z xf 00 ( x ) dx = xf ( x )- Z f ( x ) dx. Evaluating the second integral, we get Z xf 00 ( x ) dx = xf ( x )- f ( x ) + C. Taking C = 0 to evaluate the definite integral, we get Z 3 xf 00 ( x ) dx = ( xf ( x )- f ( x )) 3 = 3 f (3)- f (3) + f (0) . We must the rest of the information we know to find the values of f (3) ,f (3) and f (0). We know that the y-intercept of f is (0 ,- 8), so f (0) =- 8. The tangent line to f at x = 3 has slope f (3), so we put the equation given above in standard form, giving us y = 2 x- 4. The slope of this line is 2, which means f (3) = 2. Finally, we know that the tangent line to f at x = 3 must touch the graph of f at x = 3, so plugging in x = 3 into the equation of the tangent line gives y = 2(3)- 4 = 2. So f (3) = 2 as well. Plugging in everything we know, we find Z 3 xf 00 ( x ) dx = 3(2)- (2) + (- 8) =- 4 . 6...
View Full Document

{[ snackBarMessage ]}

Page5 / 6

We are given a function describing the rate of change of...

This preview shows document pages 5 - 6. Sign up to view the full document.

View Full Document Right Arrow Icon bookmark
Ask a homework question - tutors are online