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Subsection 2.14.2 Exercises

Exercises — Stage 1

1

What is the 180th derivative of the function \(f(x)=e^x\text{?}\)

2

Suppose \(f(x)\) is a differentiable function, with \(f'(x) \gt 0\) and \(f''(x) \gt 0\) for every \(x \in (a,b)\text{.}\) Which of the following must be true?

  1. \(f(x)\) is positive over \((a,b)\)
  2. \(f(x)\) is increasing over \((a,b)\)
  3. \(f(x)\) is increasing at a constant rate over \((a,b)\)
  4. \(f(x)\) is increasing faster and faster over \((a,b)\)
  5. \(f''''(x) \gt 0\) for some \(x \in (a,b)\)
3

Let \(f(x)=ax^{15}\) for some constant \(a\text{.}\) Which value of \(a\) results in \(f^{(15)}(x)=3\text{?}\)

4

Find the mistake(s) in the following work, and provide a corrected answer.

Suppose \(-14x^2+2xy+y^2=1\text{.}\) We find \(\ds\ddiff{2}{y}{x}\) at the point \(\left(1,3\right)\text{.}\) Differentiating implicitly:

\begin{align*} -28x+2y+2xy'+2yy'&=0\\ \end{align*}

Plugging in \(x=1\text{,}\) \(y=3\text{:}\)

\begin{align*} -28+6+2y'+6y'&=0\\ y'&=\frac{11}{4}\\ \end{align*}

Differentiating:

\begin{align*} y''&=0 \end{align*}

Exercises — Stage 2

5

Let \(f(x)=(\log x-1)x\text{.}\) Evaluate \(f''(x)\text{.}\)

6

Evaluate \(\ds\ddiff{2}{}{x}\{\arctan x\}\text{.}\)

7

The unit circle consists of all point \(x^2+y^2=1\text{.}\) Give an expression for \(\ds\ddiff{2}{y}{x}\) in terms of \(y\text{.}\)

8

Suppose the position of a particle at time \(t\) is given by \(s(t) = \dfrac{e^t}{t^2+1}\text{.}\) Find the acceleration of the particle at time \(t=1\text{.}\)

9

Evaluate \(\ds\ddiff{3}{}{x}\{\log(5x^2-12)\}\text{.}\)

10

The height of a particle at time \(t\) seconds is given by \(h(t)=-\cos t\text{.}\) Is the particle speeding up or slowing down at \(t=1\text{?}\)

11

The height of a particle at time \(t\) seconds is given by \(h(t)=t^3-t^2-5t+10\text{.}\) Is the particle's motion getting faster or slower at \(t=1\text{?}\)

12

Suppose a curve is defined implicitly by

\begin{equation*} x^2+x+y=\sin(xy) \end{equation*}

What is \(\ds\ddiff{2}{y}{x}\) at the point \((0,0)\text{?}\)

13

Which statements below are true, and which false?

  1. \(\ds\ddiff{4}{}{x} \sin x = \sin x\)
  2. \(\ds\ddiff{4}{}{x} \cos x = \cos x\)
  3. \(\ds\ddiff{4}{}{x} \tan x = \tan x\)

Exercises — Stage 3

14

A function \(f(x)\) satisfies \(f'(x) \lt 0\) and \(f''(x) \gt 0\) over \((a,b)\text{.}\) Which of the following curves below might represent \(y=f(x)\text{?}\)

15

Let \(f(x)=2^{x}\text{.}\) What is \(f^{(n)}(x)\text{,}\) if \(n\) is a whole number?

16

Let \(f(x)=ax^3+bx^2+cx+d\text{,}\) where \(a\text{,}\) \(b\text{,}\) \(c\text{,}\) and \(d\) are nonzero constants. What is the smallest integer \(n\) so that \(\ds\ddiff{n}{f}{x}=0\) for all \(x\text{?}\)

17 (✳)
\begin{equation*} f(x)=e^{x+x^2}\qquad \qquad \qquad h(x)=1+x+\frac{3}{2}x^2 \end{equation*}
  1. Find the first and second derivatives of both functions
  2. Evaluate both functions and their first and second derivatives at 0.
  3. Show that for all \(x \gt 0\text{,}\) \(f(x) \gt h(x)\text{.}\)

Remark: for some applications, we only need to know that a function is “big enough.” Since \(f(x)\) is a difficult function to evaluate, it may be useful to know that it is bigger than \(h(x)\) when \(x\) is positive.

18 (✳)

The equation \(x^3y+y^3=10x\) defines \(y\) implicitly as a function of \(x\) near the point \((1,2)\text{.}\)

  1. Compute \(y'\) at this point.
  2. It can be shown that \(y''\) is negative when \(x=1\text{.}\) Use this fact and your answer to 2.14.2.18.a to make a sketch showing the relationship of the curve to its tangent line at \((1,2)\text{.}\)
19

Let \(g(x)=f(x)e^x\text{.}\) In Question 2.7.3.12, Section 2.7, we learned that \(g'(x)=[f(x)+f'(x)]e^x\text{.}\)

  1. What is \(g''(x)\text{?}\)
  2. What is \(g'''(x)\text{?}\)
  3. Based on your answers above, guess a formula for \(g^{(4)}(x)\text{.}\) Check it by differentiating.
20

Suppose \(f(x)\) is a function whose first \(n\) derivatives exist over all real numbers, and \(f^{(n)}(x)\) has precisely \(m\) roots. What is the maximum number of roots that \(f(x)\) may have?

21

How many roots does the function \(f(x)=(x+1)\log(x+1)+\sin x - x^2\) have?

22 (✳)

Let \(f(x) = x|x|\text{.}\)

  1. Show that \(f(x)\) is differentiable at \(x = 0\text{,}\) and find \(f'(0)\text{.}\)
  2. Find the second derivative of \(f(x)\text{.}\) Explicitly state, with justification, the point(s) at which \(f''(x)\) does not exist, if any.
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