Chap 19 Review

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

Exercise 1 Which pattern-book function is the derivative of the sigmoid function pnorm()? That is, \[{\large \text{pnorm}(x)} \underset{\scriptsize \text{anti-diff}}{{\stackrel{\text{diff}}{\ \ \ \ {\Huge\rightleftharpoons}\ \ \ }}} {\LARGE ?}\]

Exercise 2 Which pattern-book function is the anti-derivative of the reciprocal \(1/x\)? That is, \[{\LARGE ?} \underset{\scriptsize \text{anti-diff}}{{\stackrel{\text{diff}}{\ \ \ \ {\Huge\rightleftharpoons}\ \ \ }}} {\large \frac{1}{x}}\]

NOTE: Differentiation produces a “child” function from a “parent” function. The child is the derivative of the parent. Putting the relationship the other way, the parent is the anti-derivative of the child. “Derivative” and “anti-derivative” are two ways of looking at the same relationship between a pair of functions. So, if \(f(x)\) is the derivative of \(F(x)\), then \(F(x)\) is the anti-derivative of \(f(x)\).

Exercise 3 Which pattern-book function is the anti-derivative of the gaussian \(\text{dnorm()}\)? That is, \[{\LARGE ?} \underset{\scriptsize \text{anti-diff}}{{\stackrel{\text{diff}}{\ \ \ \ {\Huge\rightleftharpoons}\ \ \ }}} {\large \text{dnorm}(x)}\]

NOTE: Differentiation produces a “child” function from a “parent” function. The child is the derivative of the parent. Putting the relationship the other way, the parent is the anti-derivative of the child. “Derivative” and “anti-derivative” are two ways of looking at the same relationship between a pair of functions. So, if \(f(x)\) is the derivative of \(F(x)\), then \(F(x)\) is the anti-derivative of \(f(x)\). In other words: \[{\large F(x)} \underset{\scriptsize \text{anti-diff}}{{\stackrel{\text{diff}}{\ \ \ \ {\Huge\rightleftharpoons}\ \ \ }}} {\Large f(x)}\]

Exercise 4 What is the derivative of the power-law function \(x^p\)?i That is, \[{\Large x^p} \underset{\scriptsize \text{anti-diff}}{{\stackrel{\text{diff}}{\ \ \ \ {\Huge\rightleftharpoons}\ \ \ }}} {\LARGE ?}\]

Exercise 5 There are two pattern-book functions whose second derivative is proportional to the function itself. Which are they?

Exponential and sinusoid

Exponential and sigmoid

Exponential and logarithm

Sinusoid and gaussian

question id: drill-Pattern-book-derivs-5

Exercise 6 What is the derivative of \(t^5\) with respect to \(t\)? That is, \[{\Large t^5} \underset{\scriptsize \text{anti-diff}}{{\stackrel{\text{diff}}{\ \ \ \ {\Huge\rightleftharpoons}\ \ \ }}} {\Large ?}\]

Exercise 7 What is \(\partial_x x^2\)?

Exercise 8 What is \(\partial_t \sin(x)\)

Exercise 9 Suppose you know only this one fact about \(f(x)\), that \(\partial_{xx}\, f(7.3) = 1.6\). Which of these statements must be true?

\(f(x)\) is increasing at \(x=7.3\).

\(f(x)\) is concave up and decreasing at \(x=7.3\)

\(f(x)\) is concave up at \(x=7.3\)

\(f(x)\) is concave up at \(x=7.3\), but eventually it will become concave down.

question id: drill-Pattern-book-derivs-9

Exercise 10 If \(f(x)\) is discontinuous at \(x=5\), can it possibly be smooth at \(x=6\)?

Yes No

question id: drill-Pattern-book-derivs-10

Exercise 11 If \(g(x)\) is discontinuous at \(x=1\), what will be the value of \(\partial_x g(x)\) at \(x=1\)?

Depends on how big the gap is at the discontinuity.

\(1/x\)

The derivative isn’t defined at a discontinuity.

question id: drill-Pattern-book-derivs-11

Exercise 12 Which of the following is the correct construction for \(\partial_t g(t)\)?

\(\lim_{h \rightarrow 0} \frac{g(t + h) - g(t)}{h}\)

\(\lim_{h \rightarrow 0} \frac{g(t + h) - g(t)}{t}\)

\(\lim_{h \rightarrow 0} \frac{g(t) - g(t+h)}{h}\)

\(\lim_{x \rightarrow 0} \frac{g(t + h) - g(t)}{h}\)

question id: drill-Pattern-book-derivs-12

Exercise 13 Which of these is a reasonable definition of a derivative?

A derivative is a function whose value tells, for any input, the instantaneous rate of change of the function from which it was derived.

A derivative is the slope of a function.

A derivative is a function whose value tells, for any input, the instantaneous change of the function from which it was derived.

question id: drill-Pattern-book-derivs-13

Exercise 14 Which one of these is not the derivative of a pattern-book function?

Exercise 15 Which of the following shapes of functions is not allowed? You are strongly advised to try to draw each shape.

Increasing and concave up.

Decreasing and concave up.

Increasing and concave down.

Decreasing and concave down.

None of them are allowed.

All of them are allowed.

question id: drill-Pattern-book-derivs-15

Exercise 16 If a function \(f\) is not defined at \(x=a\), then which of the following is correct?

\(\displaystyle{\lim_{x\rightarrow a} f(x)}\) cannot exist.

\(\displaystyle{\lim_{x\rightarrow a} f(x)}\) could be \(0\).

\(\displaystyle{\lim_{x\rightarrow a} f(x)}\) must approach \(\infty\)

none of the above.

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