Chap 22 Exercises

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

Exercise 1 Consider the following functions, all of which involve a domain split at \(x=0\) and the pasting together of two individually \(C^\infty\) functions:

\[f_A(x) \equiv \left\{\begin{array}{cl}{{x^2}} & \text{for}\ 0\leq x\\ x& \text{otherwise}\end{array} \right.\]

  1. How smooth is \(f_A(x)\)?
discontinuous       \(C^0\)       \(C^1\)       \(C^2\)       \(C^3\)       \(C^\infty\)      

question id: fox-knows-dress-1

\[f_B(x) \equiv \left\{\begin{array}{cl}x^3 & \text{for}\ 0 \leq x\\0&\text{otherwise}\end{array} \right.\]

  1. How smooth is \(f_B(x)\)?
discontinuous       \(C^0\)       \(C^1\)       \(C^2\)       \(C^3\)       \(C^\infty\)      

question id: fox-knows-dress-2

\[f_C(x) \equiv \left\{\begin{array}{cl}{x^3} & \text{for}\ 0 \leq x\\x^3& \text{otherwise}\end{array} \right.\]

  1. How smooth is \(f_C(x)\)?)
discontinuous       \(C^0\)       \(C^1\)       \(C^2\)       \(C^3\)       \(C^\infty\)      

question id: fox-knows-dress-3

\[f_D(x) \equiv \left\{\begin{array}{cl}x^3 & \text{for}\ 0 \leq x\\x^2&\text{otherwise}\end{array} \right.\]

  1. How smooth is \(f_D(x)\)?
discontinuous       \(C^0\)       \(C^1\)       \(C^2\)       \(C^3\)       \(C^\infty\)      

question id: fox-knows-dress-4

\[f_E(x) \equiv \left\{\begin{array}{cl}x^3 & \text{for}\ 0 \leq x\\-x^2 &\text{otherwise}\end{array} \right.\]

  1. How smooth is \(f_E(x)\)?
discontinuous       \(C^0\)       \(C^1\)       \(C^2\)       \(C^3\)       \(C^\infty\)      

question id: fox-knows-dress-5

\[f_F(x) \equiv \left\{\begin{array}{cl}\cos(x) & \text{for}\ 0 \leq x\\0 &\text{otherwise}\end{array} \right.\]

  1. How smooth is \(f_F(x)\)?
discontinuous       \(C^0\)       \(C^1\)       \(C^2\)       \(C^3\)       \(C^\infty\)      

question id: fox-knows-dress-6

\[f_G(x) \equiv \left\{\begin{array}{cl}\sin(x) & \text{for}\ 0 \leq x\\0 &\text{otherwise}\end{array} \right.\]

  1. How smooth is \(f_G(x)\)?
discontinuous       \(C^0\)       \(C^1\)       \(C^2\)       \(C^3\)       \(C^\infty\)      

question id: fox-knows-dress-7

\[f_H(x) \equiv \left\{\begin{array}{cl}\sin(x) & \text{for}\ 0 \leq x\\x &\text{otherwise}\end{array} \right.\]

  1. How smooth is \(f_H(x)\)?
discontinuous       \(C^0\)       \(C^1\)       \(C^2\)       \(C^3\)       \(C^\infty\)      

question id: fox-knows-dress-8

Exercise 2  

Figure 1
  1. In which boxes is the function shown in #fig-panda-drive-shirt-1 smooth?
A and B       B and C       A and C       none of them       all of them      

question id: panda-drive-shirt-1

Figure 2

2.In which boxes is the function in Figure fig-panda-drive-shirt-2 smooth?

A and B       B and C       A and C       none of them       all of them      

question id: panda-drive-shirt-2

  1. In which boxes is the function in @Figure fig-panda-drive-shirt-3 is smooth?
A       B       neither of them       both of them      

question id: panda-drive-shirt-3

Exercise 3 For the sketched functions below, decide what level of smoothness—\(C^0, C^1, C^2, ...\)— best describes the function. (We make no tricks in the drawings. Where a function looks like it is broken–that is, the function locally has a V-shape or a \(\Lambda\)-shape–it is meant to be broken.)

  1. What’s the smoothness level of function A(x)? (Hint: A quadratic function has a first derivative that changes with x but a second derivative that is constant for all x.)
discontinuous       \(C^0\)       \(C^1\)       \(C^2\) or higher      

question id: moose-hears-door

  1. What’s the smoothness level of function B(x)?
discontinuous       \(C^0\)       \(C^1\)       \(C^2\) or higher      

question id: moose-hears-door-2

  1. What’s the smoothness level of function C(x)?
discontinuous       \(C^0\)       \(C^1\)       \(C^2\) or higher      

question id: moose-hears-door-4

  1. What’s the smoothness level of function D(x)?
discontinuous       \(C^0\)       \(C^1\)       \(C^2\) or higher      

question id: mose-hears-door-4

Activities

Exercise 4 XREF not implemented yet explains that in differentiating a linear combination of two functions, or a product of two functions, or one function composed with another, your first task is to identify the two functions \(f()\) and \(g()\) involved. Second, compute the derivative of each of those functions on its own: \(\partial_x f(x)\) and \(\partial_x g(x)\).

Carry out these two tasks for each of the combined functions shown in the table. (The first row has been done for you as an example.)

Combination \(f()\) \(g()\) \(\partial_x f()\) \(\partial_x g()\)
\(e^x \ln(x)\) \(\ln(x)\) \(e^x\) \(\recip\) (that is \(1/x\)) \(e^x\)
\(sin(e^x)\)
\(x + x^2\)
\(1/\sin(x)\)
\(\pnorm(x)^2\)
\(\sqrt{\pnorm(x)}\)
\(\pnorm(x^2)\)
\(\pnorm(\sin(x))\)

Exercise 5 The left column in Figure fig-deer-pitch-saw shows functions A(), B(), C(), and D(). The right column shows functions dd1(), dd2(), and so on. Find which function (if any) in the right column corresponds to the 2nd derivative of a function in the left column.

Remember the concepts of “concave up” (a smile!) and “concave down” (a frown). At those values of \(x\) for which the 2nd derivative of a given function is positive, the given function will be concave up. When the 2nd derivative is negative, the given function will be concave down.

Figure 3
  1. The second derivative of Function A() is which of the following:
dd1()       dd2()       dd3()       dd4()      

question id: deer-pitch-saw-QA1

  1. The second derivative of Function B() is which of the following:
dd1()       dd2()       dd3()       dd4()      

question id: deer-pitch-saw-QA2

  1. The second derivative of Function C() is which of the following:
dd1()       dd2()       dd3()       dd4()      

question id: deer-pitch-saw-QA3

  1. The second derivative of Function D() is which of the following:
dd1()       dd2()       dd3()       dd4()      

question id: deer-pitch-saw-QA4

Exercise 6  

  1. Draw the graph of a function from your imagination over the domain \(-5 \leq x \leq 5\). The function should be continuous everywhere except at \(x = -2, 1, 3\).

  2. Draw the graph of a second function from your imagination over the same domain as in (1). The second function should be continuous everywhere in domain: no discontinuities. It should also have a derivative everywhere except at \(x = -2, 1, 3\).

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