Calculus I – Review for Test 4
Show all work in the blue book. No more than one problem per page.
1. Definitions and important theorems...
a) definition of an antiderivative
b) definition of a definite integral
c) Fundamental Theorem of Calculus - both parts
d) Definition of a Riemann Sum of f(x) on [a,b] for a partition P.
2. For the function f  x =x 3−8 on the interval [0, 6]
a. Compute the Riemann Sum using the following partition: subintervals are [0, 3]
[3, 5], and [5, 6], and w1=1, w2=3, and w3=5.
b. Compute the definite integral of f(x) from 0 to 6.
c. Compute the exact area enclosed between the graph of f(x) and the x-axis on [0, 6].
Examine the graph of f(x) for a hint on how to proceed.
3 The acceleration of a particle at time = t is given by the formula a=6 t−4 . If it is
known that v = 11 and s = 10 when t = 1:
a. find formulas for velocity (v) and position (s) of the particle.
b. find the speed, direction of motion, and postion of the particle at t = 4.
4. Compute the following integrals:
2
a.
1
b.
∫ 3  x x 3 − cos2 x  dx
∫ x sin  x 27 dx
d.
∫ 4 x 2  x 3−7 dx
f.
∫ sin x cos x dx
4
c.
5
−∫  x2 dx
2
e.
12 x 2 dx
∫  x 3−52
1
5. For f(x) = x2 - 12, use two iterations of Newton´s Method to estimate a zero of this
function. Use an initial value x0 = 6, and determine x1 and x2. Show all work!
6. Use a change of variables:
∫  x−1 x33 dx