Chap 37 Exercises

\[ \newcommand{\dnorm}{\text{dnorm}} \newcommand{\pnorm}{\text{pnorm}} \newcommand{\recip}{\text{recip}} \]

Exercise 3 Consider this function, \(f(t)\):

Assume that the “area” of each small box on the graph is the product of 1 Watt \(\times\) 1 second.

What is \[\int_1^2 f(t) dt\ \text{?}\] (Choose the closest answer. The units are in Watt-seconds.)

\[\text{Let}\ \ A = \int_1^3 f(x) dx\ \ \ \text{and}\ \ \ B = \int_2^4 f(x) dx\]

A B They are the same size

question id: seahorse-choose-canoe-2

\[\text{Let}\ \ A = \int_3^1 f(t) dt\ \ \ \text{and}\ \ \ B = \int_4^2 f(t) dt\]

Which is bigger, A or B?

A B They are the same size

question id: seahorse-choose-canoe-3

Consider the function \[g(x) \equiv \int_4^x f(t) dt\] for the next three questions

Which is bigger, \(g(8)\) or \(g(9)\)?

\(g(8)\) \(g(9)\) They are the same size Trick question! There can be no such function \(g(x)\) since \(f()\) is a function of \(t\)

question id: seahorse-choose-canoe-4

Is \(g(2.5)\) positive or negative?

positive zero negative

question id: seahorse-choose-canoe-5

At what value of \(x\) is \(g(x) = 0\)?

\[\text{Let}\ \ h(x) \equiv \int_0^x f(t) dt\]

At what value of \(x\) is \(h(x) \approx 0\)?

\(\partial_x h(x)\) is a function. When we write \(\partial_x h(3)\) we mean to evaluate that function for an input value of \(x=3\).

\(\partial_x h(3)\) \(\partial_x h(4)\) Can’t tell.

question id: seahorse-choose-canoe-8

\[\text{Let}\ \ y(x) \equiv \int_4^x f(t) dt\]

Here are four different graphs.

slice_plot(F(x) - F(4) ~ x, domain(x=0:10)) |>
  gf_labs(title = "(A)")

slice_plot(F(4) - F(x) ~ x, domain(x=0:10)) |>
  gf_labs(title = "(B)")

slice_plot(F(6) - F(x) ~ x, domain(x=0:10)) |>
  gf_labs(title = "(C)")

slice_plot(abs(F(4) - F(x)) ~ x, domain(x=0:10)) |>
  gf_labs(title = "(D)")

Which of the graphs shows \(y(x)\)?

Exercise 9 Methods such as Euler are tedious, ideal for the computer. So let’s look at some basic R functions for implementing the Euler Method when we know the function to be anti-differentiated \(f(x)\), the step size \(h\), and the domain \(a \leq x \leq b\). At the heart of the implementation is a function cumsum(), the “cumulative sum.” This does something very simple. The cumulative sum of the sequence 1, 2, 3, 4 is another sequence: 1, 3, 6, 10.

Active R chunk 2 has commands for using cumsum() to approximate the anti-derivative of a function \(f()\) over the domain \(a \leq x \leq b\).

Active R chunk 2:

question id: falcon-hurt-pot-essay

Here is a function:

g <- makeFun(exp(-0.2*(x^2))~x)

Use Active R chunk 2 to find and plot the anti-derivative of \(g(x)\) over the domain \(-6 \leq x \leq 6\).

Your task: Describe the shape of \(G(x)\). Your description can be one word from earlier in the book, if you choose it carefully. If your graph looks like a straight line, you did not appropriately change the domain above.

Exercise 6 The graphic, from The Economist news magazine, shows the number of elective treatments (e.g. minor surgery, etc.) by the British National Health Service over the period January 2019 to April 2021. The point of the graph is to show the extent to which people avoided medical treatment due to the Covid-19 pandemic, which started to become severe in March 2020. (The January/February treatment numbers for 2020 line up pretty exactly with the 2019 numbers.)

According to the graphic, how many fewer treatements were there in the 12 months of 2020 compared to the 12 months of 2019? (Choose the best of the following answers.)

Activities

Exercise 7 Using the Euler method find \(\int f(t) dt\) over the interval \(t_0=0\) to \(t_{end}=1\). The \(t\) quantity is in steps of \(h=0.01\).

\(t\)	\(\partial_t f(t)\)	\(\int f(t) dt\)
0	0.399	0.5
0.01	0.242
0.02	0.054
0.03	0.399

No answers yet collected