{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# mid1g_sol - Math 125G Spring 2002 First Mid-Term Exam...

This preview shows pages 1–2. Sign up to view the full content.

Math 125G - Spring 2002 First Mid-Term Exam Solutions 1. Is 1 2 x 2 ln x - 1 4 x 2 an antiderivative of x ln x ? Explain. The derivative of 1 2 x 2 ln x - 1 4 x 2 is x ln x (details left to the reader, must be shown for credit), so, yes, 1 2 x 2 ln x - 1 4 x 2 an antiderivative of x ln x . 2. Suppose f primeprime ( x ) = 2 + e x , f prime (0) = 3 and f (0) = 2. Find f ( x ). Since f primeprime ( x ) = 2 + e x , f prime ( x ) = integraldisplay (2 + e x ) dx = 2 x + e x + C . Using the fact that f prime (0) = 3, we have f prime (0) = 0+ e 0 + C = 1+ C = 3, so C = 2, i.e., f prime ( x ) = 2 x + e x +2. From this, we get f ( x ) = integraldisplay (2 x + e x + 2) dx = x 2 + e x + 2 x + D , where D is an arbitrary constant. Using the fact that f (0) = 2, we have 2 = f (0) = 0 + 1 + 0 + D , so D = 1. Hence, f ( x ) = 1 + 2 x + x 2 + e x . 3. Use the midpoint rule with n = 3 to approximate the integral integraldisplay 6 0 ln(sin x + 3) dx. Cutting the interval [0 , 6] into three equal subintervals, we find these subintervals have length 2 and have centers 1, 3 and 5. Hence the midpoint rule approximation is 2 (ln(sin 1 + 3) + ln(sin 3 + 3) + ln(sin 5 + 3)) = 6 . 40782354841865609 ....

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

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

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern