Review for MTH 231 Midterm on April 27th

This is a mash-up of review things I had on my computer. Some have self-grading answers, some not. We can review some on the 25th.

The test will cover: 3.4, 3.8, 3.10, 4.1, 4.2, 4.3, 4.4, 4.5, 4.6

Applications of the derivative

The derivative is interpreted in many ways:

• It is discussed in the context of rates of change in lesson 14

• In lesson 22 we learned the technique of implicit differentiation.

• In lessons 24 and 25 the rates of change are carried forward to derive equations involving rates from equations involving related variables.

• in lesson 26 the tangent line view of the derivative is used to talk about approximations: $f(x+h)\approx f(x) + f'(x)\cdot h$. This is widely employed use of the derivative.

• In lesson 27 and 28 the distinction between absolute maxima (over an interval) and local maxima is defined. There are two key theorems: (1) the extreme value theorem: for a continuous function $f(x)$ over $I=[a,b]$ there is an absolute maximum and absolute minimum (2) for a continuous function $f(x)$ over $I=[a,b]$ the absolute maximum (minimum) occurs at either a critical point or an endpoint.

• We learned the first derivative test in lessons 28, 30 to tell when a critical point will correspond to a local maximum or minimum (the derivative changes sign). This is a consequence of a simple fact: a positive derivative implies a function is increasing.

• In lessons 31 and 32 we learn – the second derivative test – a different way to tell when a critical point will correspond to a local maximum or minimum ($f''>0$ is a local min, $f'' < 0$ a local max). This test is easier to do (sometimes) but is not always conclusive. It rests on a concept of concavity that can be defined in terms of a second derivative.

• We learned in lesson 33 L'Hospital's rule, which allows us to find limits of many more expressions, and in particular gave us a tool to explore horizontal asymptotes, or behaviour in the large for an expression.

• We learned in lessons 34, 35 about how to use calculus to make informative sketches of functions

Some sample problems

• Consider the following graph

The average rate of change over $[1,2]$ is

positive

negative

zero

The average rate of change over $[2,3]$ is

positive

negative

zero

The average rate of change over $[0, 2]$ is

positive

negative

zero

The instantaneous rate of change at $2$ is

positive

negative

zero

The instantaneous rate of change at $3$ is

positive

negative

zero

• Consider the graph below over the interval $[0,2]$. Graphically identify a value $c$ with instantaneous velocity equal to the average velocity over $[0,2]$.

• An equation is given by

\[ x^2 y + x y^2 = 0 \]

Is the following true?

\[ 2xy + x^2 \frac{dy}{dx} + y^2 + 2xy\frac{dy}{dx} = 0? \]

Yes No

What is the slope of the tangent line at $(-1,1)$?

• A spy is stationed 25 feet across a street. Their eye is on a person walking along the street at 5 ft/sec. Find $d\theta/dt$ when $x=10$. (Ignore axis in graph)

• As Claude walks away from a 16 foot lampost, the tip of his shadow moves twice as fast as he does. What is Claude's height?

• If $f(x) = e^x + e -2x$, find $f''(3)$.

• Use an approximation to find $\sqrt{17}$.

• Estimate the change in $y$ for a $1$ unit change in $x$ for $f(x) = x^3 - 2x$ at $x=3$.

• In Wikipedia we find

A critical point of a function of a single real variable, $f(x)$, is a value x0 in the domain of f where it is not differentiable or its derivative is $0$ ($f′(x0) = 0$). A critical value is the image under $f$ of a critical point. These concepts may be visualized through the graph of $f$: at a critical point, the graph has a horizontal tangent if you can assign one at all.

When you have a tangent line at a critical point:

The slope is $0$

$f'(c)$ does not exist

When you have a critical point $c$ is it true that $f(c)$ is either a local maximum or local minimum?

Yes No

When you have a local maximum or minimum at $c$ is it true that $c$ is a critical point?

Yes No

• Let $f(x) = x^3 − 3x + 1$. Find the critical points of $f$.

is $-1$ a critical point?

Yes No

These solve $f'(x) = 0$ or $3x^2 - 3 = 0$

• Let $f(x) = x\cdot e^{-x}$. Find the critical points of $f$.

is $-1$ a critical point?

Yes No

You solve $f'(x)=0$ or $e^{-x}(1-x)=0$.

• Let $f(x) = x^{2/3}$. Find the one critical point of $f$.

• Let $f(x) = \sin(x) + \cos(x)$. Find the critical points of $f$ in $[0, 2\pi]$.

Is $\pi/4$ a critical point?

Yes No

• Using calculus, find the maximum and minimum value of $f(x) = \tan(x) - 2x$ on $[0,1]$.

The maximum value is

It happens at a

a critical point

an endpoint

The minimum value is

It happens at a

an endpoint

a critical point

• Using calculus, find the maximum and minimum value of $f(x) = x - 4x/(x+1)$ over $[0,3]$.

What is the minimum value over $I$?

What is the maximum value over $I$?

• Using calculus, find the minimum value of $f(x) = 10^{-16}/x + x$ for $x > 0$. (It happens at a critical point.)

• A sign chart is given for $f'(x)$:

        +    0    +   0    -     0   +
f'(x) ------ 2 ------ 4 -------- 6 ------

What are the critical points of $f(x)$? Which are local maxima? Which are local minima? Which are neither?

Is $2$ a critical point and a relative maximum?

Yes No

Is $4$ a critical point and a relative maximum?

Yes No

Is $6$ a critical point and a relative maximum?

Yes No

• From the graph estimate values of the critical points of $f(x)$

Are there critical points due to $f'(x)$ not existing?

Yes No

Are there critical points due to $f'(x)=0$?

Yes No

• From the graph, on what intervals is $f'(x) > 0$?

• Below is the graph of $f'(x)$ for some function $f(x)$. Based on the graph, what are the critical points of $f(x)$?

How many critical points are there?

Two, as we are counting the one at $0$ and the one at $2$. (The one at $0$ may not technically be a critical point, should $f$ not be defined at $2$.

From this graph, characterize the critical points as either: local maxima, local minima, or neither using the first derivative test.

• Consider the two graphs of functions below, is it possible that the second one is the first derivative of the first?

Is this $f$ and $f'$?

Yes No

• The following graph is of $f'(x)$ – and NOT $f(x)$.

What are the critical points of $f(x)$?

The values where the graph has its peaks

The values seen where the graph crosses the $x$ axis

The values where the graph has both its peaks and its valleys.

Which critical points are relative maxima of $f(x)$? Is $2$ one?

Yes No

Is $c=2.5$ among the critical points where $f$ has a relative maxima?

Yes No

On what intervals do you know $f(x)$ is increasing? Is $[1,2]$ one of them?

Yes No

On what intervals if $f(x)$ concave up? Is $[1.2]$ one of them?

Yes No

• Consider the graph of $f(x) = x^x$ over $[0,2]$ ($f(0)=1)$. Estimate the value $c$ guaranteed by the mean value theorem from the graph:

Click a value where sin(x) is increasing

---

• Why is this the wrong way to use L'Hopital's rule?

\[ \lim_{x \rightarrow 0} \frac{\cos(x)}{x} = \lim_{x\rightarrow 0} \frac{-\sin(x)}{1} = 0 \]

The denominator can be zero outside of $c$, so the rule doesn't apply

The limit is not of the form $0/0$ or $\infty/\infty$

The derivatives are wrong, you need the quotient rule

The answer is right, not wrong

• Use L'hopital's rule to find

\[ \lim_{x \rightarrow 0} \frac{\sin(5x)}{x\cos(x)} \]

• Use L'hopital's rule to find

\[ \lim_{x \rightarrow 1} \frac{2 \ln(x)}{x-1} \]

• Use L'hopital's rule to find

\[ \lim_{x \rightarrow \infty} \frac{1 - \cos(x))}{x^2} \]

How many times were needed before the resulting limit was not indeterminate?

• Use L'hopital's rule to find

\[ \lim_{x \rightarrow \infty} \frac{x^2}{e^x} \]

• You can mentally do L'Hopital's rule $n$ times to see what the result of this limit will be:

\[ \lim_{x \rightarrow \infty} \frac{x^n}{e^x} \]

What is it?

This is consistent with which statement?

Exponentials grow slower than polynomials

Exponentials and polynomials grow at the same rate

Exponentials grow faster than polynomials

True or False

• The Mean Value Theorem says for nice functions $f(x+h) - f(x) \approx f'(x) \cdot h$.

true false

No, this is the linear approximation. the mean value theorem says some $c$ exists with $f(x+h) - f(x) = f'(c) h$.

• The extreme value theorem says for $f(x)$ continuous on $[a,b]$ that there exists a $c$ in $[a,b]$ where $c$ is a critical point

true

false

No, the extreme value theorem just guarantees an absolution max and min.

• For a continuous function on $[a,b]$, we know that the absolute maximum occurs at an endpoint or at a critical point

true

false

• Rolle's theorem says that if $f$ is continous on $[a,b]$ and differentiable on $(a,b)$ and $f(a) = f(b)$, then there is some $c$ where $f'(c) = 0$.

true

false

• Rolle's theorem is a special case of the Mean Value Theorem, as the slope of the secant line is just $0$ under the assumptions.

true

false

• Consider the function $f(x) = |x|$ on $I=(-1,1)$.

This function has an absolute maximum over $I$?

Yes No

This function has an absolute minimum over $I$?

Yes No

• A function is increasing on $(a,b)$ if whenever $a < u < v < b$ that $f(u) > f(v)$.

true

false

• A function is increasing on $(a,b)$ if $f'(x) > 0$ on $(a,b)$.

true

false

• A differentiable function has a positive derivative on $(a,b)$ if it is increasing:

true

false

• A function has a derivative that changes sign from $-$ to $+$ at $x=c$. Does $c$, a critical point, correspond to a relative maximum?

Yes No

This will be a relative minimum

• A function has a derivative that changes sign from $+$ to $-$ at $x=c$. Is $c$, a critical point, a relative maximum?

Yes No

This will be a relative minimum

• A function has a derivative that does not change sign at $c$, a critical point. Is there a relative maximum or minimum at $c$?

No

Possibly, but it is not guaranteed

Yes

• At $x=c$, $f(x)$ has a critical point. It is found that $f''(c)=0$. Is $c$ possibly a relative maximum?

Yes No

Yes, a relative maximum is possible. When $f''(c)=0$ the second derivative test doesn't rule out any possibility.

• At $x=c$, $f(x)$ has a critical point. It is found that $f''(c) > 0$. Is $c$ possibly a relative maximum?

Yes No

• A function $f(x)$ is graphed below along with its transition points and the line $y=0$. (We mark points where $f$, $f'$, and f''are0`.)

Between $[a,b]$ what can you say about the first derivative?

negative

positive

undefined

zero

Between $[b,c]$ what can you say about the second derivative?

undefined

positive

negative

zero

Between $[c,d]$ what can you say about the function?

zero

positive

undefined

negative

Between $[d,e]$ what can you say about $f''(x)$?

zero

undefined

positive

negative

Between $[e,f]$ what can you say about $f'(x)$?

positive

negative

zero

undefined

Between $[f, g]$ what can you say about $f(x)$?

negative

positive

zero

undefined

• Based on the following graph of a rational function answer:

This function has:

A horizontal asymptote

two horizontal asymptotes

a slant asymptote

none of the other choices

This function has a vertical asymptote at $x=c$. Knowing $c$ is an integer, what is $c$?

This function has how many zeros?

This function has how many critical points?

Does the graph show any inflection points?

Yes No

What graph has a slant asymptote

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