1.3.1How does it change?

Finite Differences

1-96.

Upward Parabola, vertex at the origin, with dashed slope triangles between the following points, with labeled for vertical legs: (negative 3, comma 9), & (negative 2, comma 4), label negative 5, (negative 2, comma 4), & (negative 1, comma 1), label negative 3, (negative 1, comma 1), & origin, label negative 1, (1, comma 1), & (2, comma 4), label 3, (2, comma 4), & (3, comma 9), label 5. HOW DOES IT CHANGE?

A large focus in calculus is on how functions change. Whether a function is increasing or decreasing, how the change is occurring can be measured. Consider the graph of $f(x) = x^2$ . Review the graph and note how the function values are changing. Slope triangles are shown for $−3 ≤ x ≤ 3$ .

The table below shows the finite differences, $∆y$ , which are the differences between consecutive $y$ -values.

$x$	$-3$	$-2$	$-1$	$0$	$1$	$2$	$3$
$f\left(x\right)$	$9$	$4$	$1$	$0$	$1$	$4$	$9$
$\boldsymbol{\Delta\ y}$	$\boldsymbol{\underset{\LARGE\smile}{-5}}$ $\boldsymbol{\underset{\LARGE\smile}{-3}}$ $\boldsymbol{\underset{\LARGE\smile}{-1}}$ $\boldsymbol{\underset{\LARGE\smile}{+1}}$ $\boldsymbol{\underset{\LARGE\smile}{+3}}$ $\boldsymbol{\underset{\LARGE\smile}{+5}}$

Patterns among the finite differences reveal information about the way a function is changing. According to the finite differences in the table above, how are the function values changing as $x$ increases? Do you see a pattern?

1-97.

Does this pattern hold true for other quadratic functions? Using a table with finite differences labeled, describe how $Δy$ changes as $x$ increases for the functions listed below. Look for patterns within the table as well as between these different quadratic functions. Record your findings.

$f(x) = 2x^2 − 3x + 1$
$f(x) = −3x^2 + 6$

1-98.

Based on your results from problem 1-97, how do quadratic functions change? Summarize your findings by using the general equation: $f(x)=ax^2+bx+c$ . In other words, when the graph of a function is a parabola, what will the graph of its finite differences look like?

1-99.

Now consider the functions below. Describe how each of function changes as $x$ increases. Be sure to try a variety of examples to verify your observations.

Constant Functions
$f(x) = a$

Linear Functions
$f(x) = ax + b$

Cubic Functions
$f(x) = ax^3 + bx^2 + cx + d$

1-100.

Make a prediction about how the graph of $f(x) = x^n$ changes.

1-101.

Rewrite $f(x)=\left|x\right|$ as a piecewise-defined function using two linear equations. Describe how the graph grows. 1-101 HW eTool. Homework Help ✎

1-102.

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

$f(x) =\large\frac { x - 2 } { x ^ { 2 } + 4 }$
$g(x) =\large\frac { \sqrt { x + 2 } } { x ^ { 2 } + x }$

1-103.

Multiple Choice: The values of $x$ for which the graphs of $y = x + 3$ and $y^2 = 6x$ intersect are: Homework Help ✎

$−3$ and $3$
$−3$
$3$
$0$
None of these

1-104.

Determine the exact value(s) of $x$ in the domain $0 ≤ x ≤ 2π$ if: Homework Help ✎

$\sin(x) = −\frac { 1 } { 2 }$ , $\tan(x) > 0$
$\cot(x)$ is undefined, $\cos(x) > 0$
$\csc(x) = \sqrt { 2 }$ , $\sin(x) > \cos(x)$

1-105.

Given $f(x) = 2x^2 − 3$ : Homework Help ✎

Evaluate $f(2)$ .
Without writing the equation of the inverse, determine $f ^{-1}(5)$ . Explain your process.
Solve for $x$ if $f(x + 2) - f(x - 2) = 64$ .

1-106.

Using the graph of $y = f(x)$ at right, sketch the following transformations. 1-106 HW eTool Homework Help ✎

$y = −f(x)$

$y = f(x + 3)$

$y = f(x) − 2$

$y = | f ( x ) |$

Continuous linear piecewise, left piece, coming from left, horizontal to the point (negative 2, comma negative 1), turning up at the point (1, comma 2), turning down & passing through the point (3, comma 0), continuing right & down.

1-107.

Sandra is playing around with inverses and thinks she has discovered something interesting. She thinks that if $f(x) = g^{–1}(x)$ , then the areas of the regions shaded below are equal. Use $f(x) = \frac { 1 } { 3 }x + 1$ with $a = 3$ and $b = 5$ to verify Sandra’s conjecture. Homework Help ✎

1-108.

To estimate the area under a curve, rectangles are often the easiest shape to use. However, there are different ways to choose the heights of the rectangles. You have already used left endpoint and right endpoint rectangles. Another way is to use midpoint rectangles, which have heights defined at the midpoints of the intervals. For example, for the function $f(x) = \frac { 1 } { 2 }x + \cos(x)$ graphed at right, the first midpoint rectangle has a height of $f(1.5) ≈ 0.821$ .

Calculate the height of the other two rectangles then use them to approximate the area under the curve for $1 ≤ x ≤ 4$ . Homework Help ✎

First quadrant graph, x axis scaled in ones from 0 to 4, with curve of the function, f of x = 1 half x + cosine x, & 3 vertical shaded bars, bottom edges on the x axis, each with width of 1, starting at x = 1, with top edge midpoint of each bar, on the curve.

1-109.

WHICH IS BETTER? Part One

Below are different sets of rectangles to approximate the area under a curve for the same interval. Look at the three different sets of rectangles and decide which will best approximate the area under the curve of this function for $a ≤ x ≤ b$ . Homework Help ✎

Explain why your choice will determine the best approximation for the area.
Will left endpoint rectangles always be an underestimate for any function? Explain.

First quadrant graph, increasing curve opening up, starting about 1 fourth up on y axis, & 4 equal width vertical shaded bars, bottom edges on the axis, left edge of first bar labeled, a, right edge of last bar labeled, b, with top left vertex of each bar, on the curve.

Left Endpoint Rectangles

First quadrant graph, increasing curve opening up, starting about 1 fourth up on y axis, & 4 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 curve.

Midpoint Rectangles

First quadrant graph, increasing curve opening up, starting about 1 fourth up on y axis, with 4 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 top right vertex of each bar, on the curve.

Right Endpoint Rectangles

Calculus Third Edition