Chap 17 Exercises

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

Exercise 1 In this exercise, we ask you to estimate the slope from a graph of the function. But the function is exponential, so not a straight line.

A fundamental idea in calculus is that even a function with a curved graph, if you zoom in closely around a given point, will look like a straight line. And you know how to calculate the slope of a straight line.

When the graph is curved, the slope will be different at different points along the graph. So there is not a single slope for the function. Still, we can talk about the “slope at a point.”

One way to specify a point on a function’s graph is to give the horizontal coordinate: the input to the function. But here we will give you the output of the function. So long as the function passes the horizontal-line test, as the exponential does, specifying any particular output in the function’s range uniquely identifies a corresponding input.

Estimate the slope of the exponential function \(g(x) \equiv e^x\) at several inputs, which we will call \(x_1\), \(x_2\), \(x_3\) and \(x_4\). We won’t give you numerical values for the \(x_i\) points, but we will tell you the output of the function at each of those inputs. the values of \(x\) where:

  1. \(g(x_1) = 1\)
  2. \(g(x_2) = 5\)
  3. \(g(x_3) = 10\)
  4. \(g(x_4) = 0.1\)

For each of (a)-(d), use Active R chunk lst-exponentials-slope to plot the exponential function \(e^x\) on a domain zoomed in around around the appropriate value of \(x_i\). Then calculate the slope of the curve at that \(x_i\).

Active R chunk 1

Using your answers for the slopes at the points given in (a)-(d), choose the best answer to this question: What is the pattern in the slope as \(x\) varies?

The slope at each value \(x_i\) is the same as \(e^{x_i}\).

The slope at each value \(x_i\) is the same as \(x_i\).

The slope at each value of \(x_i\) is the same as \(x_i^2\).

The slope at each value of \(x\) is the same as \(\sqrt{x}.\)

question id: exponential-slopes

Exercise 2  

Glance at the graph. In which boxes is the slope negative?

A, B, C

B, C, D

A, C, D

question id: frog-bid-bed-1

Exercise 3  

Figure 1
  1. Consider the slope of the function in the domains marked by the boxes in Figure fig-turtle-send-pot. What is the order of boxes from least steep to steepest?

A, B, C

C, A, B

A, C, B

none of these

question id: turtle-send-pot-1

Exercise 4 As you know, given a function \(g(x)\) it is easy to construct a new function \({\cal D}_x g(x)\) that will be an approximation to the derivative \(\partial_x g(x)\). The approximation function, which we call the slope function, can be \[{\cal D}_x g(x) \equiv \frac{g(x + 0.1) - g(x)}{0.1}\]

In Active R chunk lst-wolf-talk-kayak, use makeFun() to create a function \(g(x) \equiv e^x\) and another function called slope_of_g() using the definition of \({\cal D} g(x)\).

Active R chunk 2
  1. What’s the value of slope_of_g(1)?
0.37       0.85       1.37       1.85       2.85      

question id: wolf-talk-kayak-1

Using your sandbox, plot both g() and slope_of_g() (in blue) on a domain \(-1 \leq x \leq 1\). This can be done with slicePlot() in the following way:

Active R chunk 3
  1. Which of these statements best describes the graph of \(g()\) compared to slope_of_g()?

slope_of_g() is negative compared to g(x).

slope_of_g() is shifted left by about \(1\) compared to g(x).

slope_of_g() has a much smaller amplitude than g().

slope_of_g() is practically the same function as g(). That is, for any input the output of the two functions is practically the same.

question id: wolf-talk-kayak-2

No answers yet collected