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Math 3A Final Exam Review

Show your work to earn full credit. Do all problems using the methods discussed in class.

1.      Estimate the area under the graph of y  = 2√x from x = 0 to x = 4 using four approximating rectangles and (a) left endpoints and (b) right endpoints. Round your answers to four decimal places.

2.      Use the limit definition of the integral to evaluate the integral.

3.     Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function and simplify.

4.      Find the general indefinite integral and simplify. (Use C for the constant of integration.)

5.      The velocity function (in meters per second) is given for a particle moving along a line.

v(t) = t2 − 3t − 15, 0 ≤ t ≤ 7

(a)   Find the displacement.

(b)  Find the distance traveled by the particle during the given time interval.

6.      Evaluate the definite integral by interpreting it as areas. Include a graph

7.      Evaluate the indefinite integral.

8.      Evaluate the definite integral.

*For problems 9 and 10. The displacement (in meters) of a particle moving in a straight line is given by s = t2  - 9t + 19, where t is measured in seconds.

9.      Find the average velocity over the specified time intervals.

(a)   [4, 4.5]

(b)  [4,4.1]

10.    Use the limit definition of a derivative to find a formula for the instantaneous velocity at any time t. Find the instantaneous velocity when t = 4.

11.  Find the derivative of the function and find all critical numbers.

f(x) = (x + 4)2 (2x − 5)2

12.    Find dy/dx by implicit differentiation.

ex/y = 8x - y

13.    A box with an open top is to be constructed from a square piece of cardboard, 3 ft. wide, by cutting out a square from each of the four corners and bending up the sides.

(a) Write an expression for the volume V in terms of x.

(b)  Find the largest volume that such a box can have.