Test 3 review

Review for test 3 in MTH229

Test 3 will be on December 18th. It will cover 2 new projects (8 and 9, but not 10) and have some questions from previous material.

As with the first two tests, you can use the computer and the internet, but you can not use your phone during the test or communicate with others or any artificial intelligence engines in any way during the test.

As with the other tests, some questions will require you to show your Julia commands to receive full credit.

As in class, we make use of the following packages:

using MTH229
using Plots

---

Here are some sample questions. Any similarity to actual questions is not intended and should not be inferred.

Interpreting the first and second derivative

Question

These graphs were produced using the plotif function from the MTH229 package.

Select the one which corresponds to plotif(f,f',a,b) for some choice of f, a, and b.

Question

Again, these graphs were produced using the plotif function from the MTH229 package.

Select the one which corresponds to plotif(f,f'',a,b) for some choice of f, a, and b.

Question

Let \(f(x) = \sin(x - \cos(x))\). This is a continuously differentiable function. Consider these commands:

f(x) = sin(x - cos(x))
fzeros(f', 0, 2pi)

2-element Vector{Float64}:
 1.5707963267948966
 4.71238898038454

What are these values mathematically?

A critical point in [a,b] for f is when the derivative is 0 or undefined. This function is continuous and has continuous derivative, so will always be defiend. So fzeros finds the critical points. Now, it is not the case that a critical point is necessarily corresponding to a relative extrema.

Now consider:

f(x) = sin(x - cos(x))
fzeros(f'', 0, 2pi)

3-element Vector{Float64}:
 0.8602451023996021
 2.281347551190191
 4.712398004144786

What are these values mathematically?

An inflection point in [a,b] for f is when the concavity changes. If, like f, the second derivative is continuous, these inflection points will typically happen when the second derivative is \(0\), it is actually required that the second derivative changes sign at the zero. So unless you check in addition, the above only finds candidates for inflection points.

Question

For some continuously differentiable function, the following sign chart is produced

sign_chart(f', -1, 1)

3-element Vector{NamedTuple{(:zero_oo_NaN, :sign_change)}}:
 (zero_oo_NaN = 0.0, sign_change = + to -)
 (zero_oo_NaN = 0.450183611295, sign_change = - to +)
 (zero_oo_NaN = 0.876726215395, sign_change = + to -)

Is the function \(f(x)\) increasing at \(x=1/2\)?

The derivative is positive around \(x=1/2\)

Does this alternate version of a sign chart indicate the same?

      [         +               0  -   0      +       0  -  ]
<-----|-------------------------|------|--------------|-----|----->
     -1                         0     0.45…          0.87…  1

It is supposed to be the same...

Question

For some continuously differentiable function, the following sign chart is produced

sign_chart(f', -1, 1)

3-element Vector{NamedTuple{(:zero_oo_NaN, :sign_change)}}:
 (zero_oo_NaN = 0.0, sign_change = + to -)
 (zero_oo_NaN = 0.450183611295, sign_change = - to +)
 (zero_oo_NaN = 0.876726215395, sign_change = + to -)

What can be said about the value 0.4501...?

A relative minimum as the function descreases on the left and increases on the right.

Question

For some continuously differentiable function, the following sign chart is produced

sign_chart(f'', -1, 1)

3-element Vector{NamedTuple{(:zero_oo_NaN, :sign_change)}}:
 (zero_oo_NaN = -0.96641123869, sign_change = + to -)
 (zero_oo_NaN = 0.196201217011, sign_change = - to +)
 (zero_oo_NaN = 0.701531539789, sign_change = + to -)

The value \(x=0.345\) is a critical point. What more can you say?

The second derivative test is used here.

Question

For some function \(f(x)\) the following code is run?

f (generic function with 1 method)

cps = fzeros(f', -1, 1)

2-element Vector{Float64}:
 -0.5235987755982989
  0.3246979402390842

f''.(cps)

2-element Vector{Float64}:
 -1.0956249566358063
  2.6831915802117856

Classify the value 0.3246979402390842.

Well, the value is a critical point, as it solve \(f'(x) = 0\). Further more \(f''(x) > 0\) so the function is concave up at this point. Hence, there is a relative minimum at this point.

Classify the value -1.0956249566358063.

Well, the value is not critical point, as it does not solve \(f'(x) = 0\).

Question

The derivative of \(f(x)\) is given by

\[ f'(x) = \cos(x - \sin^2(x)). \]

This is a plot \(f'\) over the interval \([-1,1/2]\).

Graphically identify the critical point of \(f(x)\).

At the critical point, \(f(x)\) has:

The first derivative changes sign from - to + at the critical point; by the first-derivative test this means there is a relative minimum.

Extrema

Question

Our goal: find the dimensions of the rectangle of largest area that has its base on the \(x\)-axis and its other two vertices above the x-axis and lying on the parabola \(y = 12 − x^2\).

What is the constraint for this problem?

The point \((x,y)\) lies on the parabola, so \(y = 12 - x^2\) is the constraint.

What is the objective for this problem?

The point \((x,y)\) lies on the parabola, the base of the rectangle stretches for \(-x\) to \(x\) so the area is \((2x)y\).

What is the \(x\) value for the maximum area?

Question

In an elliptical sport field we want to design a rectangular soccer field with the maximum possible area. The sport field is given by the graph of

\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1. \]

Assume \(a = 200\) and \(b=150\), find the length \(2x\) and width \(2y\) of the pitch that maximizes the area of the pitch.

What is the constraint for this problem?

One vertex of the rectangular field is \((x,y)\), a point on the ellipse.

What is the objective for this problem?

One vertex of the rectangular field is \((x,y)\), a point on the ellipse. The length of each side is \(2x\) and \(2y\), so area is \((2x)(2y)\).

What is the value for \(2x\)?

What is the value for \(2y\)?

Question

Each rectangular page of a book must contain \(30 cm^2\) of printed text, and each page must have \(2 cm\) margins at top and bottom, and \(1 cm\) margin at each side. What is the minimum possible area of such a page?

What is the constraint for this problem?

We use \(w\) and \(h\) for the size of the printed text, so \(30 = wh\) is the constraint.

What is the objective for this problem?

We use \(w\) and \(h\) for the size of the printed text, so the page size is \(w + 2(1)\) and \(h+2(2)\).

What is the minimum area?

Question

Consider all triangles formed by lines passing through the point \((x_0, y_0) = (8/9, 3)\) and both the \(x-\) and \(y\)-axes. Find the dimensions of the triangle with the shortest hypotenuse.

What is the point-slope equation of the line with slope \(m\) that goes through this point?

Google it

Let \(y_i\) be the \(y\) intercept of the line with slope \(m\) that goes through this point. What is its value?

Set \(x=0\) and simplify

Let \(x_i\) be the \(x\) intercept of the line with slope \(m\) that goes through this point. What is its value?

Set \(y=0\) and simplify

The distance formula can be used to find a function of m that expresses the hypotenuse when \(m < 0\):

d(m) = sqrt(xᵢ^2 + yᵢ^2)

Create this function.

What is the value of \(d\) when \(m=1\)?

This function has a single relative minimum near \(3\). Find the precise value.