User:Oh Isaac/R5-10,11

Problem 10: plot the local maximum and show equations
Report problem 5.10 from

Find: plot and proof

 * 1) Use MATLAB to plot $$F(5,-10;1,x)$$ near $$x=0$$ to display the local maximum (or maxima) in this region.
 * 2) Show that $$

Solution

 * {| style="width:100%" border="0"

On our honor, we did this assignment on our own, without looking at the solutions in previous semesters or other online solutions.
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
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1. As we know

in which

$$\begin{cases}(a)_0:=1\\(a)_k:=a(a+1)(a+2)\cdots(a+k-1)\end{cases}$$

The MATLAB codes for this function called is as below: ( it is not possible to get the $$k=\infty$$, so I set $$ n=k$$ as a number large enough.)

The main program is

The graph is showed below, in which $$x\in [-0.1,1]$$. And the red circle indicate the local maximum in the neighbourhood of 0.

In order to get the local maximum of $$F(a,b;c,x)$$, we can find the points of it whose differential value is 0 and make further judgment. Using the codes below to get the specific value in which the differential value is 0 in the neighbourhood of$$x=0$$.

We get the local maximum at $$x_1=0.0717,x_2=   0.2288  ,x_3=  0.4442,x_4=    0.6834 $$ near $$x=0$$, according to $$.

2. If we expand the right hand side of $$. we get

According to $$, the left hand side of $$could be written as

So the LHS=DHS.

proof end

Problem 11: plot and find
Report problem 5.11 from

Find: plot and proof

 * 1) For each value of time t, solve for altitude z(t), then plot z(t) versus t.
 * 2) Find the time when the projectile returns to the ground.

Solution

 * {| style="width:100%" border="0"

On our honor, we did this assignment on our own, without looking at the solutions in previous semesters or other online solutions.
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * }

Substitute the condition $$into the $$, we get

If we use the $$derived previously to get the equation of $$t$$, the answer and graph of it are identical.

MATLAB is unable to get the inverse function of $$z=F^{-1}(t)$$, but we can draw the plot of t when $$z(t)\in[-5,5]$$ as below with MATLAB:

So as we can see, $$t\in(-0.2,0.72)$$. If we exchange the axes of X and Y, then we get:

Unfortunately,It is hard for me to say at which point the projectile return to the ground because the physical meaning of this graph is unclear.