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Given that U is a function of x, y and z and A a vector field, prove that:
∇⋅(UA)=(∇U)⋅A+U(∇⋅A).
If A and B are vector fields, prove the following:
∇.(A×B)=B⋅(∇×A)−A⋅(∇×B).
In a journal, an article reported that the results of a peer tutoring program to help special children with disabilities learn to read. In the experiment, the children were randomly divided in two groups: the experimental group received peer tutoring along with regular instruction; and the control group received regular instruction with no peer tutoring. There were 30 children in each group. A test was given to both groups before instruction began. For experimental group, the mean score on the test was �𝑥�1� = 344.5 with sample standard deviation s1 = 49.1. For the control group, the mean score on the same test was �𝑥�2 = 354.2 with sample standard deviation s2 = 50.9. Use a 5% level of significance to test the hypothesis that there was no difference in the test of two groups before the instruction began.
A circuit has in series an electromotive force given by E=40-sin35tV, a resistor of 10Ω, and an inductor of 0.4H . If the initial current is 2 , find the current at time t>2.
The mean lifetime of 30 bulbs produced by Company A is 500 hours and the mean lifetime of 35 bulbs produced by Company B is 492 hours. If the standard deviation of all bulbs produced by Company A is 10 hours and the standard deviation of all bulbs produced by Company B is 15 hours, test at 1% significance level that the mean lifetime of bulbs produced by Company A is better than of Company B.
Evaluate F(x)=∫ √t
2 + 9
x
4
.
a) F(4) b) F’(4) c) F’’(4)
Evaluate
∫ √tanx. sec4x dx
π
4
Also integrate the definite integral using MATLAB command.
If the birth rate of a population is b(t)= − 2200e
0.024t people per year and the death rate
is d(t)= − 1460e
0.018t people per year, find the area between these curves for 0 < t < 10.
What does this area represent? Also plot the curves using MATLAB.
Let F(x)=∫
t−3
t
2+7
for − ∞ < x < ∞
x
(a) Find the value of x where F attains its minimum value.
(b) Find intervals over which F is only increasing or only decreasing.
(c) Find open intervals over which F is only concave up or only concave down.
(a) Evaluate∫[
x
√x
2+1
]dx.
(b) Use MATLAB to generate some typical integral curves of f(x) =
x
√x
2+1
over the
interval (−5,5).
