Review for test 1 in MTH229

Test 1 will be soon – March 13th!

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
• take any pictures or screenshots
• communicate with others
• Use any artificial intelligence sites 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.

As in class, we use the following two packages and ask for the Plotly backend:

using MTH229
using Plots
plotly()

[ Info: Precompiling MTH229 [ebaf19f5-aa55-568f-9bb2-d6722dad893b] (cache misses: wrong dep version loaded (4), incompatible header (14))
[ Info: Precompiling CalculusWithJuliaPlotsExt [2fd8ad5b-8953-5bb0-b347-fdafed1fb4cb] (cache misses: wrong dep version loaded (4), incompatible header (16))
[ Info: Precompiling PlotlyBase [a03496cd-edff-5a9b-9e67-9cda94a718b5] (cache misses: wrong dep version loaded (2), incompatible header (16))
┌ Warning: backend `PlotlyBase` is not installed.
└ @ Plots ~/.julia/packages/Plots/kLeqV/src/backends.jl:45
┌ Warning: backend `PlotlyKaleido` is not installed.
└ @ Plots ~/.julia/packages/Plots/kLeqV/src/backends.jl:45

Plots.PlotlyBackend()

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:

\[ \frac{1}{2a} \]

Julia code:

1 / 2a

Does the Julia code do what the math formula expects?

It is correct, but only because Julia does some wierd prioritizing of multiplication by a numeric literal. You are much better off using parentheses, as in 1/(2a).

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

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

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

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

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

Compare f(2) > sin(2)

Question

Write a Julia function to compute

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

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

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

Question

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

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

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

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, 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?

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?

a###############################################################b
a###############################b................................
a###############b................................................
........a#######b................................................
........a###b....................................................
........a#b......................................................
.........ab......................................................
         ⋮⋮                                                      


a###############################################################b
a###############################b................................
a###############b................................................
........a#######b................................................
........a###b....................................................
........a#b......................................................
.........ab......................................................
         ⋮⋮                                                      

Select an item

Different

Same

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, find the zero.

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.)