Review for test 1 in MTH229

Test 1 will be soon—March 12th.

The test will be on paper. You will have to write out all your answers and in some cases your Julia commands to receive full credit.

On the test you can use the computer and the internet, but you can not:

• use your phone during the test—it must be out of sight and out of mind
• take any pictures or screenshots
• communicate with others
• Use any AI sites/bot/agents in any way during the test.

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Here are some sample questions. Many of these are designed for immediate feedback. The actual test questions will definitely be different. The following should be helpful nonetheless.

using MTH229
using Plots
gr()

Plots.GRBackend()

Expressions

For the next several questions there is some math expression and a Julia expression. You are asked if the Julia code expresses the same calculation. Both are written as though \(a\) had some value assigned to it.

Question

Math code:

\[ \sin(\cos(a)) \]

Julia code:

sin(cos(a))

Does the Julia code do what the math formula expects?

For these functions there isn't much difference

Question

Math code:

\[ 1 + \frac{1}{2} + \frac{1}{2\cdot 3} + \frac{1}{2 \cdot 3 \cdot 4} \]

Julia code:

1 + 1/2 + 1/2*3 + 1/2*3*4

Does the Julia code do what the math formula expects?

No, the expression 1/2*3 computes as (1/2)*3 or 1.5. Not 1/6. Etc.

Question

Math code:

\[ \cos^2(a - \pi) \]

Julia code:

cos^2(a - pi)

Does the Julia code do what the math formula expects?

No, you must square the result of the call to cos, as in cos(a-pi)^2.

Question

Math code:

\[ \ln(\frac{1 - a}{a}) \]

Julia code:

ln( (1-a) / a)

Does the Julia code do what the math formula expects?

No, the function to compute natural log, \(\ln\), is just log.

Question

Math code:

\[ 1 - \frac{1 -a^2}{1 + a^2} \]

Julia code:

(1 - (1 - a^2)) / (1 + a^2)

Does the Julia code do what the math formula expects?

The first 1 should not be paired off, that is try 1 - (1-a^2)/(1 + a^2).

Question

Math code:

\[ e^{-1/2\cdot(x-1)^2} \]

Julia code:

exp^(-1/2*(x-1)^2)

Does the Julia code do what the math formula expects?

Use either exp(...) or e^(..._) (assuming the MTH229 package is loaded); this example mixes in both, so is incorrect.

Question

Which of these is actually a Julia function for some common mathematical function:

Well, hopefully you know why the others aren't, though we do use a tangent function from the MTH229 package, this is not for the common tangent function provided by tan.

Question

Suppose for specific \(a,b,\) and \(c\) values you were to solve for \(C\) (in radians) in

\[ a^2 + b^2 -2ab\cos(C) = c^2 \]

Does the following set of Julia commands compute this expression for the given values.

a, b, c = 3, 5, 6
cosC = (c^2 - (a^2 + b^2)) / (-2a*b)
acos(cosC)

Yup, that should work. It finds \(C\) in radians.

Question

Suppose for specific \(R_t\) and \(R_1\) you have to solve for \(R_2\) from:

\[ \frac{1}{R_t} = \frac{1}{R_1} + \frac{1}{R_2} \]

Does the following set of Julia commands compute this expression for the given values.

R1, Rt = 10, 8
1 / (1/Rt - 1/R1)

Yup, that should work.

Question

pemdas

Is the above image funny or informative?

It is funny and not informative – it being pretty much all wrong, but drawn in an authoritative style that lends it authority.

Functions

Question

Write a Julia function to compute

\[ f(x) = x - \frac{x^3}{6} + \frac{x^5}{120} \]

Would this function work?

f(x) = x - x^3/6 + x^5/120

That is the proper syntax: a name with arguments contained in parentheses, then an equals sign, and then a rule depending on the arguments

Use your function to identify which is greater \(f(2)\) or \(\sin(2)\)?

Compare f(2) > sin(2)

Question

Write a Julia function to compute

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

Would this function work?

f(x) = x - sin(x^2)

That is the proper syntax: a name with arguments contained in parentheses, then an equals sign, and then a rule depending on the arguments

Which value is bigger \(f(1)\), \(f(2)\), or \(f(3)\)?

Compute all 3 and observe

Question

The piece-wise defined function \(f\) is given below:

\[ f(x) = \begin{cases} 2x + 3 & x < 1 \\ 5 & x \geq 1 \end{cases} \]

It can be entered into Julia using the ternary operator by:

f(x) = x < 1 ? 2x + 3 : 5

Using this function (type it in), which values is biggest?

Compute all 3 and observe

Question

Consider the function \(f(x) = x^x\) over the interval \([1/2, 2]\). The secant line between these points is defined by

f(x) = x^x
a, b = 1/2, 2
g(x) = f(b) + (f(b) - f(a))/(b-a) * (x - b)

At the value of \(x=1\) which is bigger?

Compute both and observe

You could also have used the secant(f,a,b) function to define g.

Question

Consider the function \(f(x) = x^x\) over the interval \([1/2, 2]\). The tangent line to \(f(x)\) at \(x=1/2\) is given by

f(x) = x^x
tl = tangent(f, 1/2)

At the value of \(x=1\) which is bigger?

Compute both and observe

Plots

For these questions, you are expected to read the values from the graph to 1 decimal point.

Question

What is f(1)?

What value of x satisfies f(x)=2?

How many values of x satisfy f(x) = 3?

Question

A secant line for a function \(f\) at \(a\) and \(b\) is the line going through the two points \((a,f(a))\) and \((b, f(b))\).

A tangent line for a function \(f\) at \(c\) is the line that best approximates \(f(x)\) at the point \(x=c\).

Consider this graph:

There are two lines. What is the dashed line?

What is the dash-dotted line?

If you identified a tangent line above, what is the value of \(c\) the line is tangent at?

Question

Consider this graph of some function \(f(x)\) over \([-10,10]\):

To identify a zero of the function is impossible from the graph. Why?

To reframe the graph to try to identify a zero, which would you try?

From the graph you can see that by x=5 this graph is starting to increase so there is no zero after x=5.

Question

Consider the following plot of \(f(x) = \cos(x)\) and \(g(x) = x/10\). Where do they intersect within \([0, \pi/2]?\)

f(x) = cos(x)
g(x) = x/10
plot(f, 0, pi/2)
plot!(g)

Question

Consider the following plot of \(f(x) = \ln(x)\) and its tangent line at \(x=1/2\).

f(x) = log(x)
plot(f, 1/10, 3/2)
plot!(tangent(f, 1/2))
plot!(zero)

When does the tangent line cross the \(x\) axis?

Question

The continuous function \(f(x) = x^2 e^{-x}\) has a maximum value over the interval \([0, 4]\). What is it. (The \(y\) value)

f(x) = x^2 * exp(-x)
plot(f, 0, 4)

Question

Two lines will intersect unless they are parallel. Consider this plot of \(f(x) = x^x\) over \([1/2, 2]\) along with its secant line over \([1/2, 2]\) and its tangent line at \(x=1\).

Where do they intersect? Use Inf if they do not.

f(x)  = x^x
plot(f, 1/4, 4; ylims = (0, 4.1), legend=false)
plot!(secant(f, 1/2, 2))
plot!(tangent(f, 1))

Finding zeros

For these zero-finding problems, you can’t get enough accuracy for a correct answer just from reading the graph. You will need to use bisection or fzero.

Question

Consider \(f(x) = x^5 - x - 1\). The interval \([a_0, b_0] = [1,2]\) is a bracketing interval. From the output of bisection(f, 1, 2), you should be able to see that \([a_1, b_1] = [1, 1.5]\). What is \([a_3, b_3]?\)

\(a_3\) is equal to?

\(b_3\) is equal to?

What “value” is the zero of \(f(x)\) in \([1,2]\) guaranteed by the intermediate value theorem?

(Value in quotes, as the actual mathematical answer is only approximated by this floating point value.)

Question

Consider the following plot of some special function

plot(besselj0, -5, 5)
plot!(zero)

Which of the following intervals is a bracketing interval for besselj0?

The one where the graph crosses the x axis

Using the bracketing interval and fzero (or bisection), find the zero.

Question

For the function graphed above, which interval is not a bracketing interval? (A bracketing interval is \([a,b]\) where by the intermediate value theorem you know a zero is guaranteed to be between \(a\) and \(b\).

Note

For the remaining questions you will be using bisection or fzero to solve one of three types of equations:

• \(f(x) = 0\)
• \(f(x) = c\)
• \(f(x) = g(x)\)

We solve this with bisection or fzero, but the last two pattern types require you to introduce an auxiliary function.

Question

Consider the functions \(g(x) = \sin(x)\) and \(f(x) = \sin(x + \sin(x + \sin(x)))\). For each, \((2,3)\) is a bracketing interval. Does bisection find the exact same value or a different value when used?

Select an item

Same

Different

Question

Find a zero of the decreasing function f(x) = erfc(x) - 1/2.

(You might need to plot first to find a bracketing interval.)

Question

The function \(f(x) = \tan^{-1}(x)\) is an increasing, continuous function. Using the intermediate value theorem, this says the function will take on any value in its range exactly once. That is for any \(y\) in \((-\pi/2, \pi/2)\) there is just one \(x\) with \(f(x) = y\). Find that \(x\) value when \(y = 1.23456\)

(It is even possible for this problem to use \((-\infty, \infty)\) for the bracketing interval when using fzero.)

Question

The functions \(f(x) = xe^{-x^2}\) and \(g(x) = x + 1\) over the interval \([-2,2]\) intersect just once. Find the \(x\) value of the point of intersection using fzero.

f(x) = x * exp(-x^2)
g(x) = x + 1
plot(f, -2, 2)
plot!(g)

Question

The function f(x) = airyai(x) and the function g(x) = x/4 intersect once in the interval \([0,5]\). Where? (Use fzero.)