Math 20B, Fall Midterm Exam #1 October 25, 2002

Instructor: Professor Chow

Instructions

1. NO CALCULATOR.

2. CLOSED BOOK, CLOSED NOTES.

THESE ARE ONLY A FEW SAMPLE PROBLEMS. THE MIDTERM WILL HAVE 7 WRITTEN PROBLEMS AND 1 MULTIPLE CHOICE.

1. Let
   
   $$F(x)=\int_{3x}^{x^{2}}\frac{t^{2}}{t+1}\,dt.$$
   
   
   Calculate
   
   $$\frac{d}{dx}\left( F(x)\right) .$$
   
   
   You do not need to simplify your answer.

2. Calculate the definite integral
   
   $$\int_{1}^{e}\frac{\ln x}{x}\,dx.$$
   
   
   Hint: Use the substitution trick.

3. For the expression
   
   $$\lim_{n\rightarrow \infty }\sum_{i=1}^{n}\frac{3}{n}\left( 2+\left( \frac{3i}{n}\right) ^{2}\right)$$
   
   
   a) Determine a region whose area is equal to the given limit (Find the function $f\left( x\right)$ and the upper and lower limits $a$ and $b.$)

   b) Write it as a definite integral (DO NOT evaluate the integal).

4. Find the volume of the solid obtained by rotating the region bounded by the curves $y=x+1,$ $y=1,$ $x=1$ about the $x$-axis.