Monica

Astronomers observe a cloud which appears to be expanding. When it is observed a month later, its radius is estimated to be 5 times the original radius. After a further 3 months, the radius appears to be 5 times as large again.

It is though that the expansion is described by a differential equation of the form dr/dt=cr^m where s and m are constants. There is however, a difference of opinion about the appropriate value to take for m. Two hypotheses are proposed, that m=1/3 and m=1/2. Investigate which of these models best fits the observed data better.

... and passes through the point (1,2).

... and passes through the point (1,2).

Astronomers observe a cloud which appears to be expanding. When it is observed a month later, its radius is estimated to be 5 times the original radius. After a further 3 months, the radius appears to be 5 times as large again.

It is though that the expansion is described by a differential equation of the form dr/dt=cr^m where s and m are constants. There is however, a difference of opinion about the appropriate value to take for m. Two hypotheses are proposed, that m=1/3 and m=1/2. Investigate which of these models best fits the observed data better.

I tried illustrating the first model as ct+k=3/2*r^2/3, but I'm not sure how to find the value of k?

The rate of increase in its height, in metres per year, is given by the formula 0.2sqrt(25-h), where h is the height of the tree, in metres, t years after it is planted.

Write down a differential equation connecting h and t, and solve it to find an expression for t as a function of h?

Also, would I be right in separating the variables and solving;

int(5)dt=int(1/sqrt(25-h))dh?

A sculler is rowing a 2 kilometre course. She starts rowing at 5 metres per second, but gradually tires so that when she has rowed x metres, her speed has dropped to 5e^(-0.0001x) metres per second. How long will she take to complete the course?

A sculler is rowing a 2 kilometre course. She starts rowing at 5 metres per second, but gradually tires so that when she has rowed x metres, her speed has dropped to 5e^(-0.0001x) metres per second. How long will she take to complete the course.

dx/dt=5e^-0.0001x

dt/dx=1/5e^0.0001x

The integral of dt/dx is 2000e^0.0001x + k. However I don't get the right value for the constant k when I plug in the initial conditions?

Use substitution x=a^2/u, where a>0, to show that the integral of 1/(a^2+x^2) dx for limits [0 to a] is equal to the integral of 1/(a^2+x^2) dx for limits [infinity to a]. I'm not sure how to illustrate the change in limits using the expression x=a^2/u.

Use substitution x=a^/u, where a>0, to show that the integral of 1/(a^2+x^2) dx for limits [0 to a] is equal to the integral of 1/(a^2+x^2) dx for limits [infinity to a]. I'm not sure how to illustrate the change in limits using the expression x=a^/u.

I'm not sure how to find the new limits since u=arccos(1/x).

I was wondering if someone could walk me through the actual substitution step by step since I wind up with int((6+6tan^2u)/(4+4tan^2u))dx for the limits [1/2pi...1/2pi], which doesn't give me the right answer.

How do I go about evaluating the new limits?

How do I go about evaluating the new limits? u=arctan(x/2), but I'm not sure how plugging in x=infinity gives pi/2.

...when it it rotated about the x-axis.

int(sec^2x)dx=int(1+tan^2x)dx

=int(1)dx+int(sinx*(sinx/cos^2x))dx

How would I continue integrating the function using:

u=sinx/cosx

du=cos^-1x+2cos^-3xsin^2x dx

dv=sinx

v=-cosx

So far I've got the result x-tanx+int(1+2tan^2x)dx but I'm not sure how if it simplifies to give tanx + k and was wondering if someone could illustrate it in steps.