The Central Idea
A difference tells us how much a quantity changes.
A difference quotient compares that change with the corresponding change in input:
\[ \boxed{ \frac{\text{change in output}}{\text{change in input}} }. \]When inputs are sampled at regular steps, the output differences themselves can form useful patterns. These are called finite differences.
The two ideas are closely related:
\[ \boxed{ \text{finite difference} \longrightarrow \text{difference quotient} \longrightarrow \text{rate of change} }. \]Review: Change in Input and Output
Suppose a function changes from the point
\[ (x_1,f(x_1)) \]to the point
\[ (x_2,f(x_2)). \]The change in input is
\[ \Delta x=x_2-x_1, \]and the change in output is
\[ \Delta y=f(x_2)-f(x_1). \]The corresponding average rate of change is
\[ \boxed{ \frac{\Delta y}{\Delta x} = \frac{f(x_2)-f(x_1)}{x_2-x_1} }. \]What Is a Difference Quotient?
A difference quotient is a quotient in which a change in function value is divided by a corresponding change in input.
One common form is
\[ \boxed{ \frac{f(b)-f(a)}{b-a} }, \qquad a\ne b. \]This is the average rate of change from \(x=a\) to \(x=b\), and it is also the slope of the secant line through the corresponding points on the graph.
The \(h\)-Form of the Difference Quotient
If we begin at \(x=a\) and move by an input step \(h\), then the second input is
\[ a+h. \]The change in input is therefore
\[ (a+h)-a=h. \]The corresponding difference quotient is
\[ \boxed{ \frac{f(a+h)-f(a)}{h} }, \qquad h\ne0. \]This form is especially useful because it keeps the starting input fixed and represents the interval length with a single quantity \(h\).
Difference Quotients Are Secant Slopes
The points
\[ P=(a,f(a)) \]and
\[ Q=(a+h,f(a+h)) \]determine a secant line.
Its slope is
\[ m_{\text{sec}} = \frac{f(a+h)-f(a)}{h}. \]Therefore, the same expression can be interpreted as:
- a difference quotient;
- an average rate of change;
- a secant slope.
Example: Difference Quotient for a Linear Function
Let
\[ f(x)=3x+5. \]Then
\[ f(a+h)=3(a+h)+5. \]Expand:
\[ f(a+h)=3a+3h+5. \]Form the quotient:
\[ \frac{f(a+h)-f(a)}{h} = \frac{(3a+3h+5)-(3a+5)}{h}. \]Simplify:
\[ \frac{3h}{h} = \boxed{3}. \]The quotient is independent of \(a\) and \(h\). This matches the constant slope of the line.
Example: Difference Quotient for \(x^2\)
Let
\[ f(x)=x^2. \]Then
\[ f(a+h)=(a+h)^2. \]Expand:
\[ f(a+h)=a^2+2ah+h^2. \]Since
\[ f(a)=a^2, \]the quotient is
\[ \frac{a^2+2ah+h^2-a^2}{h}. \]Simplify:
\[ \frac{2ah+h^2}{h} = \frac{h(2a+h)}{h}. \]For \(h\ne0\),
\[ \boxed{2a+h}. \]What the Simplified Quotient Tells Us
For
\[ f(x)=x^2, \]the secant slope from \(a\) to \(a+h\) is
\[ 2a+h. \]Unlike the linear example, this value depends on location and interval size.
This reflects the changing slope of the parabola.
If \(h\) becomes very small, the term \(h\) contributes less and less to the secant slope, foreshadowing the local-rate expression \(2a\).
Why Algebraic Simplification Matters
The original expression
\[ \frac{f(a+h)-f(a)}{h} \]often appears complicated after substitution.
The goal is usually to simplify the numerator until a factor of \(h\) becomes visible.
Once that factor is extracted, it can often be canceled because the secant quotient assumes
\[ h\ne0. \]This algebraic cancellation is one of the key bridges from average change to local change.
Example: Difference Quotient for a Quadratic
Let
\[ f(x)=2x^2-3x+1. \]First compute
\[ f(a+h) = 2(a+h)^2-3(a+h)+1. \]Expand:
\[ f(a+h) = 2a^2+4ah+2h^2-3a-3h+1. \]Since
\[ f(a)=2a^2-3a+1, \]subtract:
\[ f(a+h)-f(a) = 4ah+2h^2-3h. \]Factor \(h\):
\[ h(4a+2h-3). \]Therefore,
\[ \boxed{ \frac{f(a+h)-f(a)}{h} = 4a+2h-3 }. \]Difference Quotients from Data
A formula is not required.
Suppose a table contains
| \(x\) | \(f(x)\) |
|---|---|
| \(2\) | \(7\) |
| \(3\) | \(12\) |
The difference quotient is
\[ \frac{12-7}{3-2} = \boxed{5}. \]Thus, the average output change is \(5\) units per input unit across that interval.
What Is a Finite Difference?
A finite difference is the difference between function values at distinct input values.
If the input step is \(h\), a forward finite difference can be written
\[ \boxed{ \Delta_h f(x) = f(x+h)-f(x) }. \]When the step size is understood to be \(1\), this is often shortened to
\[ \Delta f(x)=f(x+1)-f(x). \]Difference versus Difference Quotient
The finite difference
\[ f(x+h)-f(x) \]measures the raw change in output.
The difference quotient
\[ \frac{f(x+h)-f(x)}{h} \]divides that output change by the input step.
Therefore,
\[ \boxed{ \text{difference quotient} = \frac{\text{finite difference}}{\text{input step}} }. \]Why Step Size Matters
Suppose a function changes by \(12\) output units.
If the input changes by \(3\), the average rate is
\[ \frac{12}{3}=4. \]If the input changes by \(6\), the same output difference would correspond to
\[ \frac{12}{6}=2. \]A raw finite difference does not by itself tell us the rate unless the input spacing is also known.
Equally Spaced Inputs
Finite-difference tables are especially useful when input values are equally spaced.
For example,
\[ 0,\ 1,\ 2,\ 3,\ 4 \]have constant step
\[ h=1. \]Likewise,
\[ 2,\ 2.5,\ 3,\ 3.5,\ 4 \]have constant step
\[ h=0.5. \]Equal spacing lets us compare successive output changes on the same basis.
First Differences
The differences between consecutive output values are called first differences.
If the outputs are
\[ y_0,\ y_1,\ y_2,\ y_3, \]then the first differences are
\[ y_1-y_0,\qquad y_2-y_1,\qquad y_3-y_2. \]These record the change from one sampled point to the next.
Example: Constant First Differences
Consider
\[ f(x)=3x+2 \]for equally spaced integer inputs.
| \(x\) | \(f(x)\) | First difference |
|---|---|---|
| \(0\) | \(2\) | |
| \(1\) | \(5\) | \(3\) |
| \(2\) | \(8\) | \(3\) |
| \(3\) | \(11\) | \(3\) |
| \(4\) | \(14\) | \(3\) |
The first differences are constant:
\[ \boxed{3,\ 3,\ 3,\ 3}. \]Linear Functions and First Differences
For a linear function
\[ f(x)=mx+b \]with step size \(h\),
\[ f(x+h)-f(x) = m(x+h)+b-(mx+b). \]Simplifying,
\[ \boxed{ \Delta_h f(x)=mh }. \]Thus, equally spaced inputs produce constant first differences.
Dividing by the step \(h\) gives
\[ \frac{mh}{h}=m, \]the constant slope.
Second Differences
When first differences are not constant, we can take differences again.
The differences between consecutive first differences are called second differences.
Symbolically,
\[ \Delta_h^2 f(x) = \Delta_h(\Delta_h f(x)). \]Expanding this definition gives
\[ \boxed{ \Delta_h^2 f(x) = f(x+2h)-2f(x+h)+f(x) }. \]Example: Quadratic Finite Differences
Consider
\[ f(x)=x^2. \]| \(x\) | \(f(x)\) | First difference | Second difference |
|---|---|---|---|
| \(0\) | \(0\) | ||
| \(1\) | \(1\) | \(1\) | |
| \(2\) | \(4\) | \(3\) | \(2\) |
| \(3\) | \(9\) | \(5\) | \(2\) |
| \(4\) | \(16\) | \(7\) | \(2\) |
The first differences change, but the second differences are constant:
\[ \boxed{2,\ 2,\ 2}. \]Why Quadratics Have Constant Second Differences
Consider
\[ f(x)=ax^2+bx+c. \]With a fixed step \(h\), the first difference is
\[ f(x+h)-f(x). \]After simplification,
\[ \Delta_h f(x) = 2ahx+ah^2+bh. \]This is linear in \(x\).
Taking one more finite difference gives
\[ \boxed{ \Delta_h^2 f(x)=2ah^2 }. \]Because \(a\) and \(h\) are fixed, the second difference is constant.
Step Size and Quadratic Second Differences
For
\[ f(x)=ax^2+bx+c, \]the constant second difference is
\[ 2ah^2. \]This means the numerical value of the second difference depends on the input spacing.
If \(h=1\),
\[ \Delta^2 f=2a. \]If \(h=2\),
\[ \Delta_2^2 f=8a. \]The polynomial has not changed; the sampling interval has.
Third Differences
We can continue taking differences.
Differences of second differences are called third differences:
\[ \Delta_h^3f(x) = \Delta_h(\Delta_h^2f(x)). \]Cubic polynomials have constant third differences when sampled at equally spaced inputs.
Example: Cubic Finite Differences
Consider
\[ f(x)=x^3. \]| \(x\) | \(f(x)\) | First | Second | Third |
|---|---|---|---|---|
| \(0\) | \(0\) | |||
| \(1\) | \(1\) | \(1\) | ||
| \(2\) | \(8\) | \(7\) | \(6\) | |
| \(3\) | \(27\) | \(19\) | \(12\) | \(6\) |
| \(4\) | \(64\) | \(37\) | \(18\) | \(6\) |
| \(5\) | \(125\) | \(61\) | \(24\) | \(6\) |
The third differences are constant:
\[ \boxed{6,\ 6,\ 6}. \]Polynomial Degree and Finite Differences
For a polynomial of degree \(n\) with nonzero leading coefficient, equally spaced inputs produce constant \(n\)th finite differences.
| Polynomial type | Degree | First constant difference level |
|---|---|---|
| Constant | \(0\) | Function values themselves are constant |
| Linear | \(1\) | First differences |
| Quadratic | \(2\) | Second differences |
| Cubic | \(3\) | Third differences |
This pattern provides a useful way to recognize polynomial behavior from tabulated data.
Finite Differences as a Degree Detector
Suppose equally spaced data produce first differences
\[ 4,\ 7,\ 10,\ 13. \]These are not constant.
Their second differences are
\[ 3,\ 3,\ 3. \]Constant second differences are consistent with a quadratic pattern.
This does not by itself prove that every underlying real-world process is exactly quadratic, but it is strong evidence that a quadratic model may fit the sampled values.
Example: Recognizing a Quadratic Table
| \(x\) | \(y\) | First difference | Second difference |
|---|---|---|---|
| \(0\) | \(1\) | ||
| \(1\) | \(4\) | \(3\) | |
| \(2\) | \(9\) | \(5\) | \(2\) |
| \(3\) | \(16\) | \(7\) | \(2\) |
| \(4\) | \(25\) | \(9\) | \(2\) |
The constant second differences indicate quadratic behavior.
In fact, these values come from
\[ y=(x+1)^2. \]Finite Differences Can Extend a Pattern
Suppose a sequence of outputs is
\[ 2,\ 5,\ 10,\ 17,\ 26. \]The first differences are
\[ 3,\ 5,\ 7,\ 9. \]The second differences are
\[ 2,\ 2,\ 2. \]If we assume the quadratic pattern continues, the next first difference is
\[ 9+2=11. \]Therefore, the next output is
\[ 26+11=\boxed{37}. \]Forward Differences
The expression
\[ \Delta_h f(x)=f(x+h)-f(x) \]is called a forward difference because it compares the current input \(x\) with the later input \(x+h\).
Dividing by \(h\) gives the forward difference quotient
\[ \boxed{ \frac{f(x+h)-f(x)}{h} }. \]This gives the secant slope across the interval \([x,x+h]\) when \(h>0\).
Backward Differences
We may instead compare the current input with an earlier one.
A backward difference is
\[ f(x)-f(x-h). \]The corresponding backward difference quotient is
\[ \boxed{ \frac{f(x)-f(x-h)}{h} }. \]This measures average change across the interval from \(x-h\) to \(x\).
Centered Differences
A centered difference compares values on opposite sides of \(x\):
\[ f(x+h)-f(x-h). \]The total input separation is
\[ 2h. \]Therefore, the centered difference quotient is
\[ \boxed{ \frac{f(x+h)-f(x-h)}{2h} }. \]This is the secant slope through two points symmetrically placed around \(x\).
Example: Forward, Backward, and Centered Estimates
Let
\[ f(x)=x^2 \]and estimate the local rate near \(x=2\) using \(h=0.1\).
The forward quotient is
\[ \frac{f(2.1)-f(2)}{0.1} = \frac{4.41-4}{0.1} = 4.1. \]The backward quotient is
\[ \frac{f(2)-f(1.9)}{0.1} = \frac{4-3.61}{0.1} = 3.9. \]The centered quotient is
\[ \frac{f(2.1)-f(1.9)}{0.2} = \frac{4.41-3.61}{0.2} = \boxed{4}. \]Why Centered Differences Can Be Useful
A forward difference looks only to one side of the target input. A backward difference looks only to the other side.
A centered difference uses information from both sides.
For smooth functions and small equal steps, this often gives a useful numerical estimate of the local slope.
Later numerical-analysis methods make this idea more precise.
Finite Differences and Local Rate
A finite difference always uses a nonzero input separation.
Therefore, a finite-difference quotient is fundamentally an average rate across a finite interval.
However, if the interval becomes smaller and the quotient approaches a stable value, that value can represent the local rate.
\[ \boxed{ \text{small-interval average rates} \longrightarrow \text{local rate} }. \]Finite Does Not Mean Approximate
The word finite means that the input separation is nonzero and has a definite size.
A finite difference such as
\[ f(3)-f(2) \]can be computed exactly when the function values are exact.
Approximation enters when a finite difference quotient is used to estimate a local rate or when the data themselves are measured or rounded.
Difference Tables and Discrete Change
Difference tables are naturally suited to discrete data.
If a quantity is recorded at times
\[ 0,\ 1,\ 2,\ 3,\ldots, \]the first differences describe how much the quantity changes during each time step.
The second differences describe how those first changes themselves change.
This creates a discrete analogue of repeatedly studying rates of change.
Example: Position Data
Suppose an object's position, in meters, is recorded once per second.
| Time \(t\) | Position \(s(t)\) | First difference |
|---|---|---|
| \(0\) | \(0\) | |
| \(1\) | \(3\) | \(3\) |
| \(2\) | \(8\) | \(5\) |
| \(3\) | \(15\) | \(7\) |
| \(4\) | \(24\) | \(9\) |
Because each time step is \(1\) second, these first differences are also average velocities in meters per second across the corresponding one-second intervals.
Second Differences in Motion Data
The first differences in the previous example are
\[ 3,\ 5,\ 7,\ 9. \]Their second differences are
\[ 2,\ 2,\ 2. \]Thus, the average velocity increases by the same amount each second.
This is a discrete signal of constant acceleration-like behavior.
Later calculus lessons will connect this pattern to successive derivatives.
Units in Finite Differences
A finite difference has the same units as the function output.
If \(s(t)\) is measured in meters, then
\[ s(t+h)-s(t) \]is measured in meters.
The difference quotient
\[ \frac{s(t+h)-s(t)}{h} \]has units of meters per unit of time.
Tracking units helps distinguish raw change from rate of change.
Unequally Spaced Data
Ordinary finite-difference pattern tests assume equal input spacing.
Suppose the inputs are
\[ 0,\ 1,\ 3,\ 6. \]The step sizes are
\[ 1,\ 2,\ 3, \]so raw consecutive output differences are not directly comparable as equal interval changes.
In such a case, difference quotients are usually more informative because each output difference is divided by its corresponding input difference.
Example: Why Equal Spacing Matters
For
\[ f(x)=2x, \]consider the inputs
\[ 0,\ 1,\ 3. \]The outputs are
\[ 0,\ 2,\ 6. \]The raw differences are
\[ 2,\ 4. \]They are not equal because the input steps are \(1\) and \(2\).
But the difference quotients are
\[ \frac{2}{1}=2 \]and
\[ \frac{4}{2}=2. \]The constant rate is recovered once the unequal step sizes are accounted for.
Finite Differences and Polynomial Interpolation
A finite table of equally spaced values can often be modeled by a polynomial.
Constant finite differences help identify the likely polynomial degree and can support interpolation between or extension beyond known values.
This idea is useful in numerical computation because polynomials are easy to evaluate and manipulate.
More advanced treatments build entire interpolation formulas from finite differences.
Repeated Differences Reduce Polynomial Degree
Taking a finite difference of a nonconstant polynomial lowers its degree by one.
For example:
- a cubic polynomial produces a quadratic first-difference expression;
- that quadratic produces a linear second-difference expression;
- that linear expression produces a constant third difference;
- one more difference produces zero.
This mirrors, at a discrete level, a pattern that will later appear with derivatives.
Example: Algebraic First Difference of \(x^3\)
Let
\[ f(x)=x^3 \]with step \(h=1\).
Then
\[ \Delta f(x) = f(x+1)-f(x). \]Compute:
\[ (x+1)^3-x^3 = x^3+3x^2+3x+1-x^3. \]Therefore,
\[ \boxed{ \Delta f(x)=3x^2+3x+1 }. \]A cubic has produced a quadratic first-difference expression.
Second and Third Differences of \(x^3\)
Starting from
\[ \Delta f(x)=3x^2+3x+1, \]take another unit-step difference:
\[ \Delta^2 f(x) = [3(x+1)^2+3(x+1)+1] - [3x^2+3x+1]. \]Simplifying gives
\[ \Delta^2 f(x)=6x+6. \]One more difference gives
\[ \boxed{ \Delta^3 f(x)=6 }. \]This matches the constant third differences seen in the table.
Finite Difference Notation
The symbol
\[ \Delta \]is the capital Greek letter delta and is commonly used to indicate a finite change.
Thus,
\[ \Delta x \]means a finite change in \(x\), while
\[ \Delta f \]can represent a finite change in the function value.
Repeated notation such as
\[ \Delta^2f \]means a difference of differences, not the square of the original function.
Finite Difference versus Differential Notation
It is useful to keep several symbols conceptually separate.
- \(\Delta x\) usually represents a finite change in \(x\).
- \(\Delta f\) represents a corresponding finite change in function value.
- \(\dfrac{\Delta f}{\Delta x}\) represents an average rate across a finite interval.
- Derivative notation such as \(f'(x)\) represents local rate when the derivative exists.
Later calculus develops additional differential notation, but the distinction between finite change and local rate should remain clear.
Difference Quotients and the Derivative
The forward difference quotient
\[ \frac{f(x+h)-f(x)}{h} \]describes average rate over a nonzero interval of width \(h\).
If these quotients approach a unique value as \(h\to0\), that value is the derivative:
\[ \boxed{ f'(x) = \lim_{h\to0} \frac{f(x+h)-f(x)}{h} } \]when the limit exists.
The next lesson will begin using this connection to find derivatives of simple polynomials.
Common Error: Reversing Only One Difference
In a quotient such as
\[ \frac{f(b)-f(a)}{b-a}, \]the order in the numerator and denominator must match.
If the numerator is reversed, the denominator must also be reversed:
\[ \frac{f(a)-f(b)}{a-b}. \]These two quotients are equal because both numerator and denominator change sign.
Common Error: Forgetting to Divide by the Input Step
Suppose
\[ f(5)-f(2)=12. \]The number \(12\) is the output difference.
The average rate is
\[ \frac{12}{5-2} = \boxed{4}. \]Raw change and rate of change are related but not identical quantities.
Common Error: Setting \(h=0\) in the Difference Quotient
The expression
\[ \frac{f(x+h)-f(x)}{h} \]requires
\[ h\ne0. \]Setting \(h=0\) directly gives division by zero.
In derivative work, we simplify for nonzero \(h\) and then examine what the quotient approaches as \(h\) tends toward zero.
Common Error: Canceling \(h\) Across Addition
Consider
\[ \frac{2a+h}{h}. \]The \(h\) cannot be canceled from only one term of the numerator.
Cancellation applies to common factors, not to terms joined by addition or subtraction.
By contrast,
\[ \frac{h(2a+h)}{h} \]has a common factor of \(h\), so for \(h\ne0\),
\[ \frac{h(2a+h)}{h}=2a+h. \]Common Error: Ignoring Input Spacing in a Difference Table
Constant raw first differences indicate a linear pattern only when the inputs are equally spaced.
If input intervals vary, the output differences should be normalized by their corresponding input changes.
In that setting, compare
\[ \frac{\Delta y}{\Delta x} \]rather than \(\Delta y\) alone.
Common Error: Calling Every Constant Difference a First Difference
A quadratic typically has changing first differences but constant second differences.
A cubic typically has changing first and second differences but constant third differences.
Always identify the level at which the differences become constant.
Common Error: Assuming Constant Second Differences Prove a Physical Law
Constant second differences in equally spaced data are consistent with a quadratic model.
But measured data may contain rounding, noise, limited sample size, or changing conditions.
Difference patterns help identify mathematical structure; they do not by themselves establish the cause of a real-world process.
A Reliable Procedure for Simplifying a Difference Quotient
- Write the function \(f(x)\) clearly.
- Replace every \(x\) in the formula with \(x+h\) to find \(f(x+h)\).
- Form \(f(x+h)-f(x)\) using parentheses carefully.
- Expand and combine like terms.
- Factor the numerator when possible.
- Cancel only common factors.
- Remember that the original quotient assumes \(h\ne0\).
- Interpret the simplified expression as an interval rate or secant slope.
A Reliable Procedure for Building a Difference Table
- Verify that the input values are equally spaced.
- List the function or data values in order.
- Subtract each output from the next to form first differences.
- If the first differences are not constant, difference them again.
- Continue to higher differences if useful.
- Record the first level at which the differences become constant.
- Use that level to recognize possible polynomial degree.
- Keep the input step size in mind when interpreting numerical values.
Example: Full Difference-Quotient Workflow
Let
\[ f(x)=x^2-4x+7. \]Find and simplify
\[ \frac{f(x+h)-f(x)}{h}. \]First,
\[ f(x+h) = (x+h)^2-4(x+h)+7. \]Expand:
\[ f(x+h) = x^2+2xh+h^2-4x-4h+7. \]Subtract \(f(x)\):
\[ f(x+h)-f(x) = x^2+2xh+h^2-4x-4h+7 - (x^2-4x+7). \]Simplify:
\[ 2xh+h^2-4h. \]Factor:
\[ h(2x+h-4). \]Divide by \(h\):
\[ \boxed{ \frac{f(x+h)-f(x)}{h} = 2x+h-4 }. \]Example: Full Finite-Difference Workflow
Consider the outputs
\[ 3,\ 8,\ 15,\ 24,\ 35 \]at inputs
\[ 0,\ 1,\ 2,\ 3,\ 4. \]First differences:
\[ 5,\ 7,\ 9,\ 11. \]Second differences:
\[ 2,\ 2,\ 2. \]The constant second differences indicate a quadratic pattern.
If the pattern continues, the next first difference is
\[ 11+2=13. \]The next output is
\[ 35+13=\boxed{48}. \]Practice 1: Basic Difference Quotient
Let
\[ f(x)=5x-2. \]Simplify
\[ \frac{f(x+h)-f(x)}{h}. \]Practice 2: Quadratic Difference Quotient
Let
\[ f(x)=x^2+3x. \]Simplify
\[ \frac{f(x+h)-f(x)}{h}. \]Practice 3: Evaluate a Difference Quotient
For
\[ f(x)=x^2, \]compute the average rate of change from \(x=4\) to \(x=4.5\).
Practice 4: First Differences
Find the first differences of
\[ 4,\ 9,\ 14,\ 19,\ 24. \]What type of polynomial pattern is suggested?
Practice 5: Second Differences
For the outputs
\[ 1,\ 4,\ 9,\ 16,\ 25, \]find the first and second differences.
Practice 6: Extend a Quadratic Pattern
The outputs are
\[ 6,\ 11,\ 18,\ 27,\ 38. \]Assuming the constant-second-difference pattern continues, find the next output.
Practice 7: Identify Polynomial Degree
Equally spaced data have third differences that are constant and nonzero, while the second differences are not constant.
What polynomial degree is suggested?
Practice 8: Unequal Spacing
A linear function produces the following sampled values:
| \(x\) | \(f(x)\) |
|---|---|
| \(0\) | \(1\) |
| \(2\) | \(7\) |
| \(5\) | \(16\) |
The raw output differences are not equal. Show that the difference quotients are equal.
Practice 9: Centered Difference
For
\[ f(x)=x^2 \]at \(x=3\), use \(h=0.1\) to compute
\[ \frac{f(3.1)-f(2.9)}{0.2}. \]Practice 10: Identify the Error
A student sees outputs
\[ 2,\ 6,\ 14 \]at inputs
\[ 0,\ 1,\ 3 \]and concludes that the pattern is nonlinear because the output differences are \(4\) and \(8\).
Explain why that conclusion is not justified from the raw differences alone.
Practice Answers
Practice 1
\[ f(x+h)=5(x+h)-2=5x+5h-2. \] \[ \frac{f(x+h)-f(x)}{h} = \frac{5h}{h} = \boxed{5}. \]Practice 2
\[ f(x+h) = (x+h)^2+3(x+h) \] \[ = x^2+2xh+h^2+3x+3h. \]Therefore,
\[ \frac{f(x+h)-f(x)}{h} = \frac{2xh+h^2+3h}{h}. \] \[ = \boxed{2x+h+3}. \]Practice 3
\[ f(4)=16, \qquad f(4.5)=20.25. \] \[ \frac{20.25-16}{4.5-4} = \frac{4.25}{0.5} = \boxed{8.5}. \]Practice 4
The first differences are
\[ \boxed{5,\ 5,\ 5,\ 5}. \]Constant first differences suggest a linear pattern.
Practice 5
First differences:
\[ 3,\ 5,\ 7,\ 9. \]Second differences:
\[ \boxed{2,\ 2,\ 2}. \]Practice 6
First differences:
\[ 5,\ 7,\ 9,\ 11. \]Second differences:
\[ 2,\ 2,\ 2. \]The next first difference is
\[ 13, \]so the next output is
\[ 38+13=\boxed{51}. \]Practice 7
Constant nonzero third differences suggest a
\[ \boxed{\text{cubic polynomial}} \]under the equally spaced polynomial-pattern assumption.
Practice 8
From \(x=0\) to \(x=2\),
\[ \frac{7-1}{2-0} = \frac{6}{2} = 3. \]From \(x=2\) to \(x=5\),
\[ \frac{16-7}{5-2} = \frac{9}{3} = 3. \]Thus, both intervals have rate
\[ \boxed{3}. \]Practice 9
\[ f(3.1)=9.61, \qquad f(2.9)=8.41. \] \[ \frac{9.61-8.41}{0.2} = \frac{1.20}{0.2} = \boxed{6}. \]Practice 10
The input spacing is not equal:
\[ 1-0=1, \qquad 3-1=2. \]Therefore, the raw output differences \(4\) and \(8\) should not be compared as though they came from equal intervals.
The corresponding difference quotients are
\[ \frac{4}{1}=4 \]and
\[ \frac{8}{2}=4. \]The data are actually consistent with a constant rate of
\[ \boxed{4}. \]What to Remember
A finite difference measures output change:
\[ \boxed{ \Delta_hf(x)=f(x+h)-f(x) }. \]A difference quotient converts that finite change into a rate:
\[ \boxed{ \frac{f(x+h)-f(x)}{h} }. \]- difference quotients are secant slopes and average rates of change;
- the input step \(h\) must be nonzero in a finite difference quotient;
- equally spaced inputs are essential for ordinary finite-difference pattern tests;
- linear functions have constant first differences;
- quadratic functions have constant second differences;
- cubic functions have constant third differences;
- more generally, degree-\(n\) polynomials have constant \(n\)th differences for equally spaced inputs;
- forward, backward, and centered quotients provide different ways to measure nearby change;
- shrinking finite intervals leads naturally toward the derivative.
Where We Go Next
Difference quotients give an algebraic expression for secant slope, and finite differences reveal how polynomial change behaves across discrete steps. The next step is to let the secant interval shrink and extract a rule for local change.
The next lesson, Derivatives of Simple Polynomials, develops the derivative from these ideas and begins computing local rates for constant, linear, quadratic, and other simple polynomial functions.