Difference Quotients

A difference quotient is a mathematical expression that is commonly used to establish an average rate of change between two points belonging to a smoothly varying function.

Suppose $a$a and $b$b are two values in the domain of a function $f$f. The function values at $a$a and $b$b are $f(a)$f(a) and $f(b)$f(b). The difference between $a$a and $b$b, given by $b-a$b−a, corresponds to the difference between the function values, given by $f(b)-f(a)$f(b)−f(a). 

The difference quotient $\frac{f(b)-f(a)}{b-a}$f(b)−f(a)b−a​ expresses the relative change of the function value for the given change in the domain variable.

When this is pictured in the form of a graph, we may speak of this as the rise over the run, meaning the change in the function, given by the vertical difference between the points, for a given change in the domain variable, given by the horizontal difference between the points.

Equivalently, we can say that the difference quotient expresses the gradient of the line joining the points $\left(a,f(a)\right)$(a,f(a)) and $\left(b,f(b)\right)$(b,f(b)).

 

The difference quotient expresses the average rate of change of a function in moving from, say, $x=a$x=a to $x=b$x=b. When $b$b is close to $a$a, the difference quotient is near in value to the instantaneous rate of change of the function at $a$a. On a graph, we see that the secant line joining $\left(a,f(a)\right)$(a,f(a)) and $\left(b,f(b)\right)$(b,f(b)) becomes indistinguishable from the tangent line at $a$a when $b$b moves very close to $a$a.

We say that the gradient of the tangent at $a$a is the limit as $b\rightarrow a$b→a of the difference quotient $\frac{f(b)-f(a)}{b-a}$f(b)−f(a)b−a​. The gradient of the tangent is the instantaneous rate of change of the function at $a$a.

 

To make more general statements about difference quotients and gradients, we consider a point $x$x in the domain of a function and another point $x+h$x+h that is close to $x$x when $h$h is small. The difference quotient now takes the form $\frac{f(x+h)-f(x)}{x+h-x}$f(x+h)−f(x)x+h−x​, which is just

$\frac{f(x+h)-f(x)}{h}$f(x+h)−f(x)h​

Observe that if we let $h=0$h=0, the quotient becomes $\frac{0}{0}$00​, which is meaningless. And yet, if we let $h$h approach $0$0 ever more closely, we often see that the difference quotient approaches a definite value. This discovery is the basis for differential calculus.

 

Example 1

Let $y$y be a function of $x$x where $x$x can be any real number. The function is given by the formula $y(x)=\frac{x}{2}+3$y(x)=x2​+3. Find an expression for the difference quotient at any point $x$x in the domain.

According to the function definition, we have $y(x+h)=\frac{x+h}{2}+3$y(x+h)=x+h2​+3. So, we can form the difference quotient 

$\frac{y(x+h)-y(x)}{h}$y(x+h)−y(x)h​	$=$=	$\frac{\frac{x+h}{2}+3-(\frac{x}{2}+3)}{h}$x+h2​+3−(x2​+3)h​
	$=$=	$\frac{\frac{x}{2}+\frac{h}{2}+3-\frac{x}{2}-3}{h}$x2​+h2​+3−x2​−3h​
	$=$=	$\frac{\frac{h}{2}}{h}$h2​h​
	$=$=	$\frac{1}{2}$12​

We see that the gradient of a line joining any two points belonging to this function is always $\frac{1}{2}$12​, as we would expect for a function whose graph is a straight line.

 

Example 2

Find the general difference quotient expression for the function $f(x)=x^2-x+2$f(x)=x2−x+2. Use it to find the gradients of secant lines at $x=1$x=1 when $h=1,0.1,0.01,0.001$h=1,0.1,0.01,0.001.

We have, $f(x+h)=(x+h)^2-(x+h)+2$f(x+h)=(x+h)2−(x+h)+2. Therefore

$\frac{f(x+h)-f(x)}{h}$f(x+h)−f(x)h​	$=$=	$\frac{(x+h)^2-(x+h)+2-(x^2-x+2)}{h}$(x+h)2−(x+h)+2−(x2−x+2)h​
	$=$=	$\frac{x^2+2xh+h^2-x-h+2-x^2+x-2}{h}$x2+2xh+h2−x−h+2−x2+x−2h​
	$=$=	$\frac{2xh+h^2-h}{h}$2xh+h2−hh​
	$=$=	$\frac{h(2x+h-1)}{h}$h(2x+h−1)h​
	$=$=	$2x+h-1$2x+h−1

The secant lines at $x=1$x=1, therefore, have gradients $1+h$1+h. Thus, when $h=1$h=1 the secant line has gradient $2$2; when $h=0.1$h=0.1, the gradient is $1.1$1.1; when $h=0.01$h=0.01, the gradient is $1.01$1.01 and when $h=0.001$h=0.001, the secant line has gradient $001$001.

Worked Examples

Question 1

question 2

question 3