3.3.4Do you want to play a derivative game?

Ways to Describe $f^{\prime}$ and $f^{\prime\prime}$

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Math Notes

Derivative Notations

In calculus, there is not a single correct notation for expressing a derivative. Instead, several different notations have been used by different mathematicians. The usefulness of each notation varies in a given context. The most common derivative notations are listed below.

Derivative functions describe the instantaneous rate of change of another function at any value of $x$ . Since a derivative describes a rate, it is often expressed as a fraction.

First derivative notation:

$\large\frac { d y } { d x }$ read as “ $dy, dx$ ” or “ $dy$ over $dx$ ”

$f^{\prime}(x)$ read as “ $f$ prime of $x$ ”

Second derivative notation:

$\large\frac { d ^ { 2 } y } { d x ^ { 2 } }$ (or $\large\frac { d ( d y / d x ) } { d x }$ ) read as “the second derivative of $y$ with respect to $x$ ”

$f^{\prime \prime}(x)$ read as “ $f$ double prime of $x$ ”

Higher derivatives are expressed as:

$\large\frac { d ^ { n } y } { d x ^ { n } }$ read as “the $n^{\text{th}}$ derivative of $y$ with respect to $x$ ”

$f^{(n)}(x)$ read as “the $n^{\text{th}}$ derivative of $f$ with respect to $x$ ”

3-124.

THE SILENT BOARD GAME

Do not talk! Your teacher will draw a large curve on the board. It is your job to label it as completely as possible.

3-125.

WALK–A-WAVE CHALLENGE

Your teacher will give you a limited amount of time to “walk” one cycle of a perfect sine wave and record it on a motion detector. Use $y=\sin(x)+2$ . Pay attention to concavity as you walk. When should you move quickly? When should you move slowly? The team with the best graph wins!

3-126.

THE SECOND DERIVATIVE IN MOTION PROBLEMS

If $f^\prime$ represents the rate of change of $f$ , then what does $f^{\prime \prime}$ represent? If $f^\prime$ represents velocity, then what does $f^{\prime \prime}$ represent?
Since concavity depends on how the slope is changing, concavity must depend on the slope of the slope. What does this mean? Explain this in your own words.

Examine the curves below and complete the table with the signs (positive or negative) of $f^\prime$ and $f^{\prime \prime}$ . The first entry has been done for you.

$\boldsymbol {f(x)}$	Increasing or Decreasing? Walking away from the motion detector? or Walking towards the motion detector?	Concave Up or Concave Down? Getting faster? or Getting slower?	$\boldsymbol {f^\prime(x)}$	$\boldsymbol {f^{\prime \prime }(x)}$
	Increasing Away from motion detector	Concave up Getting faster	positive	positive

How does the increasing or decreasing nature of the graph of $y = f(x)$ relate to $f^\prime(x)$ ? How about $f^{\prime \prime }(x)$ ?
How does the concavity of the graph of $y = f(x)$ relate to $f^{\prime \prime}(x)$ ? How about $f^\prime(x)$ ?
In Lesson 1.4.4, you discovered that positive acceleration does not always mean that an object is speeding up, and negative acceleration does not always mean that an object is slowing down. Use the table in part (c) to determine how the graphs of $y = f^\prime(x)$ and $y = f^{\prime \prime}(x)$ can be used collectively to determine if a moving object is speeding up or slowing down. Explain your answer graphically. Be prepared to share your answer with the class.

3-127.

Apply your conclusions from parts (d) and (e) of problem 3-126 to determine where $f(x) = x^3 − 12x$ is increasing and decreasing. Also, determine the intervals over which this function is concave up and concave down.

3-128.

The Math Notes box in Lesson 3.3.2 states that a point of inflection is a point where concavity changes.

Examine the graph of $y = x^3 − 12x$ and identify where the graph changes concavity. Where is the point of inflection?
What is special about $y^{\prime \prime}$ at the point of inflection?
When locating a point of inflection, it is often helpful to set the second derivative equal to zero, and then solve for $x$ . Does this always work?

Find a counterexample to the claim that inflection points exist where the second derivative has a root. That is, can you think of a function, $f$ , such that, for some value
$x = a, f^{\prime \prime}(a) = 0,$ but $x = a$ is not the location of a point of inflection?

3-129.

Thoroughly investigate the graph of $f(x) = x^3 − 12x + 4$ . Identify all of the important qualities, such as where the function is increasing, decreasing, concave up, and concave down. Also identify any special point(s) and intercepts and provide graphs of $f^\prime$ and $f^{\prime \prime}$ . Justify all statements graphically and analytically.

3-130.

Summarize your understanding of first and second derivatives. Include information regarding increasing, decreasing, concavity.

3-131.

The graph of $y = f(x)$ is shown at right. Use the graph to list the following values in ascending (increasing) order: 3-131 HW eTool. Homework Help ✎

$0, f^\prime(1), f ^\prime(4),\frac { f ( 3 ) - f ( 1 ) } { 3 - 1 }$

First quadrant, increasing curve, opening down, starting at the origin, passing through the approximate points, (1, comma 2), (3, comma 3), (4, comma 3.5), continuing up & right.

3-132.

Without your calculator, write the equation of the line tangent to $g ( z ) = \large\frac { z ^ { 7 } + 5 z ^ { 6 } - z ^ { 3 } } { z ^ { 2 } }$ at $z = 1$ . Homework Help ✎

Compute without a calculator

3-133.

If $f^\prime (x) = −6x^{1/2} − \sin(x)$ , write a possible function for $f$ . Then write another possible equation. Homework Help ✎

3-134.

For each of the following functions, write the second derivative with respect to $x$ . Homework Help ✎

$y = 8x^{99}$

$y = −3\sqrt { x }$

$f(x)=\frac{2}{3}x-6x^{-2}$

$f(x)=7−2\cos(x)$

3-135.

If a function $f$ is increasing, such as the one at right, what must be true about $f^\prime$ ? Homework Help ✎

Explain your idea using the graph at right.
Use your idea to algebraically determine where $g(x) = 3x^2 − 3x + 1$ is increasing.

Continuous increasing curve, starting in the lower left, changing concavity in third quadrant, passing through the origin, changing concavity in first quadrant, continuing up & right.

3-136.

Write a Riemann sum to estimate the area under the cuve for $2\le x\le3$ using $20$ left endpoint rectangles given $f(x)=9x−2$ . Then compute the actual area geometrically and calculate the percent error. Homework Help ✎

3-137.

For the absolute value function shown at right, something interesting happens to the tangent line of the function at $x=2$ . Can you draw more than one tangent line at $x=2$ ? Do you think this function has any valid tangent lines at $x=2$ ? Homework Help ✎

Upward V graph, vertex at the point (2, comma 1), passing through the point (4 comma 3).

3-138.

For each graph below: Homework Help ✎

Trace the graph onto your paper and write a slope statement for $f$ .
Sketch the graph of $y = f ^\prime (x)$ using a different color.

3-139.

The velocity of a roller coaster car in meters per second on a certain segment of track is represented by the function $v(t) = -8t$ . Homework Help ✎

Sketch a graph of the velocity function for $0\le t\le4$ . What is the speed of the car at $t = 4$ ?
Assuming that the car is $100$ meters above ground at $t = 0$ , where is the car at $t = 4$ ? How far is this drop?

3-140.

Evaluate each limit. If the limit does not exist due to a vertical asymptote, then add an approach statement stating if $y$ is approaching negative or positive infinity. Homework Help ✎

$\lim\limits _ { x \rightarrow - 2 } ( 2 x ^ { 2 } - 6 x + 5 ) ^ { 2 }$

$\lim\limits _ { x \rightarrow \infty } \large\frac { e ^ { x } + e ^ { - x } } { e ^ { x } - e ^ { - x } }$

$\lim\limits _ { x \rightarrow 2 ^ { - } } \large \frac { ( x - 1 ) ( x - 2 ) } { x + 1 }$

$\lim\limits _ { x \rightarrow 2 ^ { - } } \large \frac { x + 2 } { ( x - 1 ) ( x - 2 ) }$

Calculus Third Edition

Derivative Notations