3.2.1What is a derivative?

Definition of a Derivative

How does a graphing calculator determine the slope of a tangent line? Or, how did mathematicians determine slopes before technology was available? In this lesson, we will look at slopes of secant lines and tangent lines by examining three different methods to determine the exact slope of a tangent line at its point of tangency.

To start, we will revisit the use of secant lines by studying the different slopes when $x = 4$ in the Ramp Lab. Later, we will use secant lines to help us determine the slopes of tangent lines with precision.

3-35.

Each of the following students used a different method to estimate the velocity of the ball at $t = 4$ seconds.

$\boldsymbol x$ (seconds)	$1$	$2$	$3$	$4$	$5$	$6$
$\boldsymbol {f(x)}$ (meters)	$0.2$	$0.8$	$1.8$	$3.2$	$5.0$	$7.2$

Hana estimated $f ^ { \prime } ( 4 ) \approx m \approx \frac { f ( 5 ) - f ( 4 ) } { 5 - 4 } = \frac { 5.0 - 3.2 } { 5 - 4 } = 1.8$ .

Anah estimated the slope as $f ^ { \prime } ( 4 ) \approx m \approx \frac { f ( 4 ) - f ( 3 ) } { 4 - 3 } = \frac { 3.2 - 1.8 } { 4 - 3 } = 1.4$ .

Hanah estimated the slope as $f ^ { \prime } ( 4 ) \approx m \approx \frac { f ( 5 ) - f ( 3 ) } { 5 - 3 } = \frac { 5.0 - 1.8 } { 5 - 3 } = 1.6$ .

What are the units of the slopes found by each student? Explain why.
Compare and contrast each student’s method.
Trace the graph at right of $y = f(x)$ on your paper and sketch the secant lines that Hana, Anah, and Hanah used to approximate the slope at $t = 4$ .
Is Hanah’s slope the average of Hana and Anah’s? Does this mean that Hanah’s method gives the best approximation?

First quadrant upward increasing curve, passing through the following, highlighted approximate points, (3, comma 2), (4, comma 3), (5, comma 5).

Explore this using the 3-35d Student eTool. Click in the lower right corner of the graph to view it in full-screen mode.

3-36.

To approximate the slope of the curve $f(x) = 0.25x^2$ at $x = 2$ , Devin picked another point on the curve, a small distance of $h$ to the right of $x = 2$ .

If the second point is located at the point $(2 + h, f(2 + h))$ , write and simplify an expression that represents the slope of this secant line.
Whose method did Devin use from problem 3-35: Hana’s, Anah’s, or Hanah’s?
To get the exact slope of the tangent line at $x = 2$ , what should be done with $h$ ? Write an expression that represents the true slope at $x = 2$ .
Use your expression from part (c) to determine the slope of the curve at $x = 2$ .
Does it matter if $h$ is positive? Why or why not?

First quadrant upward increasing curve, 2 tick marks labeled, 2 &, 2 + h, on the x axis, with highlighted points on the curve, corresponding to tick marks, & dashed slope triangle on the curve, between highlighted points, with increasing line, passing through the highlighted points.

3-37.

Hanah wants to use her method to determine the slope $f^\prime (1)$ for $f(x) = 4 − x^2$ . Hanah picks two points equidistant from $x = 1$ but very close to $x = 1$ .

Write and simplify an expression to represent the slope of Hanah’s secant line and then calculate the slope of the tangent line.
Use Hanah’s method on $g(x) = x^2 − 11x + 2$ to determine the slope of the tangent line when $x = 3$ .
Write a slope function that represents the slope of $g$ for all values of $x$ .

3-38.

Use each method from problem 3-35 to write general algebraic expressions which approximate the slope of a function $f$ at any point $x$ using increments of size $h$ , where $h > 0$ . For each expression, provide a sketch of a generic function $f$ and the secant line being represented.

3-39.

The Math Notes box in this lesson features the definition of the derivative using Hana’s method. Write the definition of the derivative using Anah’s method and the definition of the derivative at a point $(x = a)$ using Hanah’s method.

3-40.

Use the definition of the derivative to write a slope function, for $f(x) = 4x^2 − 3$ . Then use your slope function to calculate $f^\prime (11)$ and $f^\prime (1000)$ .

3-41.

Lulu used the limit below to write the derivative of a function $f$ .

$f^\prime (x) = \lim\limits _ { h \rightarrow 0 } \frac { 2 ( x + h ) ^ { 3 } - 2 x ^ { 3 } } { h }$

What is the equation of $f$ ?
What is the equation of $f^\prime$ ?

3-42.

Rewrite each of the following expressions in the form of $x^n$ . Homework Help ✎

$\sqrt { x }$

$\large\frac { 1 } { x ^ { 3 } }$

$\sqrt { \frac { 1 } { x } }$

$x \cdot \sqrt [ 3 ] { x ^ { 2 } }$

3-43.

Write and evaluate a Riemann sum to estimate the area under the curve for $−1\le x\le1$ given $f(x) = x^4 − x^2$ . Choose the number of rectangles so that your answer will be a good approximation of the area. What is interesting about the sign of your result? Use a description or sketch of the function to support your answer. Homework Help ✎

3-44.

Given the function $f ( x ) = \large\frac { x ^ { 2 } + 2 x - 8 } { x ^ { 2 } - 2 x }$ , calculate the following limits without using your graphing calculator. Homework Help ✎

Compute without a calculator

$\lim\limits _ { x \rightarrow 6 } f ( x )$

$\lim\limits _ { x \rightarrow 2 } f ( x )$

$\lim\limits _ { x \rightarrow \infty } f ( x )$

$\lim\limits _ { x \rightarrow 0 } f ( x )$

3-45.

At right is the graph of a function $f$ with tangent lines drawn at $x = −2, −1, 1,$ and $2$ . $f ^\prime(0) = 0$ . Use the slopes provided in the graph to write the slope function $f^\prime$ . Also, write an equation for $f$ . Homework Help ✎

Upward parabola, vertex at the origin, with 4 highlighted points, each with line tangent to the curve at the point, & labeled as follows: at x = negative 2, label is m = negative 8, at x = negative 1, label is, m = negative 4, at x = 1, label is m = 4, at x = 2, label is m = 8.

3-46.

Calculate the area under the curve for $0\le x\le8$ given $h ( x ) = -\left| x - 4 \right| + 4$ . Sketch the graph and shade the region. Homework Help ✎

3-47.

After completing the Ramp Lab in Chapter 2, Marquita decided to ride her bicycle down a nearby hill to gain more data on rolling objects. The data she collected is shown in the table below. Homework Help ✎

Time (sec)	$0.0$	$2.0$	$3.1$	$6.0$	$7.5$	$9.0$
Distance (m)	$0.0$	$3.8$	$9.1$	$32.7$	$49.7$	$69.5$

Can Marquita use this table to determine her exact velocity at $t = 6$ Explain.
Approximate Marquita’s velocity at $t = 6$ using Hana’s method.
Marquita’s teacher observed that her data fits the function $d(t) = 0.789t^2 + 0.703t − 0.338$ . Assuming that her teacher is correct, compute Marquita’s exact velocities at $t = 6$ and $t = 10$ .

3-48.

Write a slope statement to describe the graph below. Homework Help ✎

Downward curve, coming from third quadrant, passing through the positive y axis, turning down & changing to opening up, in the first quadrant, turning up also in first quadrant, continuing up & right.

3-49.

Write the equation of a tangent line that is a linear approximation to the function $y=\tan(x)$ at $x =\frac { \pi } { 4 }$ . Using your graphing calculator, investigate over what $x$ -values this approximation is reasonable. Record your observations. Homework Help ✎

3-50.

WHAT A DAY!

Below is a graph of the distance David traveled away from his home on a trip to the mountains. Place the events listed at right in the proper order based on details from the graph.

Trace the graph below and identify the parts that correspond to each event during David’s trip.

Then answer parts (a) through (d) below. Homework Help ✎

EVENTS

David has to drive back to pick up his credit card that he forgot at the restaurant.
David’s car breaks down and he is towed back to a repair shop near his house.
David stops for gas and gets a quick bite to eat.
David gets pulled over and receives a speeding ticket. He then continues his trip at a slower rate.

What are the units for the slope of the curve?
What is significant about the slope of the curve when David is stopped?
How does the slope of the curve tell you when David is speeding?
Interpret the graph where the slope is negative. What is David doing then?

3-51.

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 ✎

$\lim\limits _ { x \rightarrow - 3 ^ { + } } \frac { x - 3 } { x + 3 }$

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

$\lim\limits _{ x \rightarrow \infty } \frac { 2 + 2 ^ { x } } { 2 - 2 ^ { x } }$

$\lim\limits _ { x \rightarrow \pi ^ { + } } \pi$ (Careful!)

Calculus Third Edition

3.2.1What is a derivative?

The Derivative