Test 3 review

Review for test 3 in MTH229

Test 3 will be on May 22nd. It will cover the last 3 projects.

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 in any way during the test.

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

As in class, we use the following packages:

using MTH229
using Plots

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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

Let \(f(x) = \sin(x - \cos(x))\). Consider these commands:

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

2-element Vector{Float64}:
 1.5707963267948966
 4.71238898038454

What do they do 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 defined. So fzeros finds the critical points. Now, it is not the case that a critical point is necessarily corresponding to a relative extrema. That is why the first- and second-derivative tests are taught.

Now consider the Julia commands:

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

3-element Vector{Float64}:
 0.8602451023996021
 2.281347551190191
 4.712398004144786

What do they do 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 function, the following sign chart is produced

f (generic function with 1 method)

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 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 solves \(f'(x) = 0\). 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 a critical point, as it does not solve \(f'(x) = 0\). (It is just a \(y\) value at some critical point.)

Question

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

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

Plot \(f'\) over the interval \([-1,1/2]\). Graphically identify the lone critical point of \(f(x)\) in this interval.

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?

A figure to illustrate

Question

In an elliptical sport field we want to design a rectangular 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?

Integration

Question

Let \(f(x) = \sin(x^2)\) What is the right-Riemann sum with \(n=10\) for \(\int_0^{\pi/4} f(x) dx\)?

If \(n=10\) and the left-Riemann sum is chosen, what is the value?

If \(n=10\) and Simpson’s method is used for riemann, what is the value?

Using quadgk what is found for the definite integral?

Question

It is known that \(\int_0^\pi \sin(x) dx = 2\). Does a right Riemann sum have 5 digits of accuracy when \(n=100\)?

Question

Based on the plot of \(f(x) = \cos(x)\) and \(g(x) = 1 - x^2/2\) which integral is larger?

@syms x
plot(cos(x), -sqrt(2), sqrt(2); label="cos")
plot!(1-x^2/2;                  label = "1-x^2/2")

The graph shows (just barely) that \(\cos(x) \geq 1 - x^2/2 \geq 0\) on this interval, so the integrals have similar inequalities, as they represent areas under a curve.

Question

An ellipse is given by

\[ \frac{x^2}{200^2} + \frac{y^2}{100^2} = 1 \]

Find the perimeter.

Hint: Solve for \(y\) using the non-negative solution to the square-root; find the arc-length for \(x \in [-200, 200]\); double to get your answer. The arc-length of a function \(f(x)\) over an interval \([a,b]\) is given by the integral \(\int_a^b \sqrt{1 + f'(x)^2}\).

Question

The function \(r(x) = (1 - x^4)^{(1/3)}\) between \([-1, 1]\) is rotated around the \(x\) axis. What is the resulting volume?

Hint: use \(V = \int_a^b \pi r(x)^2\), where \(r(x)\) is the radius describing the rotation.