1.3.2How can I describe a changing graph?

Slope Statements and Finite Differences of Non-Polynomials

1-110.

The path of the roller coaster is shown below.
Your teacher will provide you with a model.

Describe the path of the roller coaster so that someone who has not seen it can draw it. Be sure to include words that will help to describe the steepness of the curve as well as its direction.
When writing a slope statement, it is reasonable to start at the left of the graph and move right—just like you read a sentence in English. Make a list of words that are useful when describing the path of a graph.

1-111.

The following two slope statements describe the same graph. Read both statements. Then sketch a graph of the function described.

“The graph starts off flat at the left and starts to increase at $x = -3$ until the graph flattens out at $x = 0$ . Then the values decreases until the graph flattens out around $x = 2$ and continues to stay flat.”

“The graph starts off flat at the left but slowly gets steeper. The slope starts getting really steep at $x = -2$ , but at $x = -1$ , the slope becomes less steep. At $x = 0$ , the slope is flat for an instant and then gets steeper but negative. At $x = 1$ , the slope starts to become less steep again, eventually getting closer and closer to zero slope.”

1-112.

Finite differences can be used to analyze the slope of a graph at various $x$ -values. Some graphs have predictable slope patterns. For example, in Lesson 1.3.1, you found patterns in the way polynomial functions change. For example, cubic functions change with a quadratic pattern, quadratic functions change with a linear pattern, and linear functions change with a constant pattern. What about other functions?

Your team will be assigned one of the function groups listed below to investigate. For each of the two functions in your function group, complete the following tasks:

Graph the function.
State the domain and range using appropriate notation.
Analyze the finite differences.
Write a slope statement.

Function Group	Equation (a)	Equation (b)
Rational	$f (x) = \frac { 1 } { x }$	$f (x) =\frac { 1 } { x ^ { 2 } }$
Trigonometric	$f (x) = \sin(x)$	$f (x) = \cos(x)$
Exponential	$f (x) = (0.5)^x$	$f (x) = 2^x$
Logarithmic	$f (x) = \log(x)$	$f (x) = \log_2(x)$
Radical	$f (x) = \sqrt { x }$	$f (x) =\sqrt [ 3 ] { x }$

1-113.

Write a piecewise-defined function that will generate the graph at right. Homework Help ✎

First quadrant continuous piecewise, starting at (0, comma 6), increasing curve opening down, turning at (2, comma 8), continuing to decrease & open down to (5, comma 3.5), turning, increasing, opening down passing through highlighted point (9, comma 6), continuing up & right.

1-114.

State the domain of each of the following functions. Homework Help ✎

$f(x) =\sqrt { 25 - x ^ { 2 } }$
$g(x) = \log(x + 5)$
$h(x) =\frac { 5 x } { x ^ { 2 } - x - 12 }$
$k(x) =\frac { \sqrt { x + 2 } } { x ^ { 2 } - 4 }$

1-115.

Simplify: $( ( \frac { x ^ { - 1 } + x ^ { 2 } } { x } ) - x + x ^ { - 2 } ) ^ { - 2 }$ Homework Help ✎

1-116.

Calculus problems often require using one or more of the trigonometric identities to solve problems. Solve each of the following equations on the interval $[0, 2π)$ . Use exact values. Homework Help ✎

$\tan(x) · \csc(x) = 2$

$\sin(x) · \cos(x) =\frac { 1 } { 4 }$

$2\sin^2(x) - \cos(x) - 1 = 0$

$\tan(x) + \cot(x) = -2$

1-117.

For each part below, give an example of a function with specified attributes. Provide a sketch of each function. Homework Help ✎

A function with a hole at $x = 3$ and an asymptote at $x = −1$ .
A function with asymptotes at the $y$ -axis and $x = 5$ and a hole at $x = −4$ .
A function with an end-behavior function $g(x) = 3x − 1$ .

1-118.

Some of the basic functions have special qualities that you have investigated in this chapter. 1-118 HW eTool. Homework Help ✎

Sketch $y = \sin(x)$ on your paper. Darken in the largest portion of the graph containing $x = 0$ for which the function passes both the horizontal and vertical line tests. State the restricted domain and range for this portion of the graph.
We use the darkened portion of the graph to sketch $y = \sin^{–1}(x)$ , making sure it is a function. Then state the domain and range.
Repeat parts (a) and (b) for $y = \cos(x)$ .

1-119.

A function $g$ is even. What can you conclude about the inverse of $g$ ? Explain. Homework Help ✎

1-120.

A flag in the shape of a quarter-circle is shown at right. Homework Help ✎

Horizontal segment, with shaded top right quarter of a circle, above the segment, with its radius, labeled 5, on the right third of the segment.

Imagine rotating the flag about its pole and describe the resulting three-dimensional figure. Draw a picture of this figure on your paper. Try this using the 1-120 eTool

Calculate the volume of the rotated flag.

1-121.

WHICH IS BETTER? Part Two

Below is a comparison between using rectangles and trapezoids to approximate the area under a curve for the same interval of a function. Decide which method you think will best approximate the area under the curve for $a ≤ x ≤ b$ . Then approximate the area using each method if $f(x) = -0.25x(x - 9)$ , $a = 2$ , and $b = 8$ using $3$ sections. Compare your results with the actual area $A = 25.5 \text{ un}^2$ . Homework Help ✎

First quadrant graph, downward parabola, vertex in center of quadrant, with point at the origin, 3 equal width, vertical shaded bars, bottom edges on x axis, left edge of first bar labeled a, right edge of last bar labeled b, & with midpoint of top edge of each bar, on the parabola.

Midpoint Rectangles

First quadrant graph, downward parabola, vertex in center of quadrant, with point at the origin, 3 equal width vertical shaded trapezoids, bottom edges on x axis, left edge of first bar labeled a, right edge of last bar labeled b, & top right & left vertices of each bar is on the parabola, so each top edge is slanted, connecting consecutive x integer points on the parabola.

Trapezoids

Calculus Third Edition

1.3.2How can I describe a changing graph?

Common Trigonometric Identities