Calculus I Fall 1999   REVIEW PROBLEMS FOR FINAL EXAM

(Note: exponentiation indicated by a "^", so the square of x is x^2, and 1 over this is x^{-2}.)

The following problems do not constitute a "practice final". They are chosen to help you
identify topics you need to review.

1. State whether each of the following functions is (i) continuous at x=0 (ii) differentiable
at x=0. Justify your answers:

(a) y=1/(1+x^2), (b) y= 1/|x| if x not zero, y(0)=1, (c) y=x^{2/3}.

2. Sketch the curve y=2x+ x^{-2}, -2 < x< 2.
Indicate local maxima or minima, regions where the curve
is rising, falling, concave up, concave down.

3. The line tangent to the curve y=1-2x-3x^2+4x^3 at the point (0,1) intersects the
y-axis where?

4. Find the following derivatives:

(a) dy/dx if y= 2^{cos(x)} (b) dy/dx if y=(1+tan^2x)^{1/2}

(c) dy/dx if y=tan^{-1}(x^2) (d) #10 on page 230 (e) # 20 on page 230

(e) #30 on page 238 (f)#38 page 246.

5. The sides of a rectangle are changing with time. The length if increasing at the rate
of 7 feet per second, and the width is decreasing at the rate of 3 feet per second. When the
length is 12 feet and the width is 5 feet fincd the rate of change of

(a) the area of the rectangle
(b) the perimeter
(c) the length of a diagonal.


6. Using a linear approximation, compute approximately (2.97)^6, given that 3^6=729.

7. #52, page 285.

8. State the mean value theorem. Show that the theorem applies to y=1/(1+x), 1 <= x <= 2,
by finding the the point c.

9. Pages 311-312, ##6,8,18,50.

10.Page 335, #28

11. Antidifferentiation: pages 356-357: 36, 40.

12. Problems involving  integration:

 (a) #36 page 389 (b) ##20,32,36 page 399 (c) ##22,26 page 407 (d) ##48,52 page 417.