Why Endpoint Slopes Need a One-Sided Approach
At a point inside an interval, a nearby point on the graph can approach from either side. At an endpoint, one of those sides may lie outside the function’s domain. A tangent-slope estimate must use only points where the function is defined.
The available secants still provide useful information. As their second point moves toward the endpoint, their slopes may approach a finite value. That value is the endpoint’s one-sided limiting slope. It describes the graph from within the interval; it is not automatically a two-sided derivative.
Secants Approaching From Inside the Interval
Suppose \(f\) is defined on a closed interval \([a,b]\), where \(a<b\). The graph includes the endpoint \((a,f(a))\). To form a secant through this point and a nearby point in the domain, choose \(x>a\). Its slope is
The available points approach \(a\) from the right, so the relevant limit is \(x\to a^+\). Equivalently, set \(x=a+h\), where \(h>0\), and let \(h\to0^+\). At the right endpoint \(b\), points in the interval approach from the left. One convenient form is \(x=b-h\), again with \(h>0\).
The sign in the right-endpoint denominator matters: the horizontal change from \(b\) to \(b-h\) is \(-h\), not \(h\). The subscripts \(+\) and \(-\) indicate the side from which the input approaches the endpoint.
Worked Example: Secants at a Left Endpoint
Worked Example: Secants at a Left Endpoint
Let \(f(x)=x^2\) on \([0,3]\). Estimate the limiting slope at the left endpoint \(x=0\).
A nearby point in the domain has the form \(x=h\), with \(h>0\). The endpoint is \((0,f(0))=(0,0)\), and the secant slope is
For example, if \(h=1\), the slope is \(1\). If \(h=0.1\), the slope is \(0.1\). If \(h=0.01\), the slope is \(0.01\). As \(h\to0^+\), these slopes approach \(0\), so \(f'_+(0)=0\).
The limiting line through the endpoint with slope \(0\) is \(y=0\). The secant calculation checks the result directly: for every \(h>0\), its slope is exactly \(h\), and \(h\to0\). Thus the secant slopes converge to the slope of that horizontal line.
Worked Example: Secants at a Right Endpoint
Worked Example: Secants at a Right Endpoint
Let \(f(x)=x^2-3x\) on \([1,3]\). Find the limiting slope at the right endpoint \(x=3\) using points from inside the interval.
First calculate the endpoint value:
Choose a distance \(h>0\) and approach from the left using \(x=3-h\). The secant’s horizontal change from the endpoint is \((3-h)-3=-h\). Its vertical change is
Therefore its slope is
For \(h=0.5\), the slope is \(2.5\); for \(h=0.1\), it is \(2.9\); and for \(h=0.01\), it is \(2.99\). As \(h\to0^+\), the slopes approach \(3\), so \(f'_-(3)=3\). Substitution into the exact secant formula verifies this: each slope is \(3-h\), whose limit is \(3\).
Why the One-Sided Limit Gives the Endpoint Slope
Each secant in these examples passes through the endpoint and a second point of the graph. If its slope is \(m\), its line has equation \(y-f(a)=m(x-a)\) at a left endpoint. When the secant slopes approach a finite number \(L\), the line equations approach \(y-f(a)=L(x-a)\): the fixed endpoint remains the same, and only the slope changes toward \(L\). This is the geometric reason that a finite limiting secant slope supplies the endpoint’s tangent-slope estimate.
This argument uses the endpoint value \(f(a)\) or \(f(b)\) and the limit from within the interval. It does not require function values beyond the domain. The limit must exist and be finite for the endpoint slope described here to be a real number.
One-Sided Endpoint Slope Is Not Necessarily a Two-Sided Derivative
The ordinary two-sided derivative at an interior point requires a function to be defined on both sides near that point and requires the difference quotient to have the same limit from both sides. At a domain endpoint, the function may have no values on the other side at all. Consequently, an endpoint slope calculated from the domain is one-sided.
Some courses use the word “derivative” at an endpoint to mean the appropriate one-sided derivative. Other contexts reserve “derivative” for the two-sided limit and call the endpoint quantity a one-sided derivative. Stating the side explicitly avoids ambiguity.
Worked Example: The Domain Determines Which Limit Is Available
Worked Example: The Domain Determines Which Limit Is Available
Consider \(f(x)=x\) on \([0,2]\). At the left endpoint, \(f(0)=0\). For \(h>0\), the secant slope from \(0\) to \(h\) is
Thus the right-hand endpoint slope is \(f'_+(0)=1\). This calculation uses only points in \([0,2]\).
Now compare two possible extensions to the left of \(0\). The extension \(F(x)=x\) for all real \(x\) has left-side quotient
Its left and right quotients both approach \(1\). Instead, define \(G(x)=2x\) for \(x<0\), while keeping \(G(x)=x\) for \(x\geq0\). Its left-side quotient is
while its right-side quotient remains \(1\). The two-sided limit for this extension therefore does not exist, since its left and right limits are \(2\) and \(1\). Yet the original function on \([0,2]\) still has the same right-hand endpoint slope \(1\). The values outside the domain are not part of that one-sided calculation.
A Common Mistake: Using a Point Outside the Interval
At a right endpoint, using \(b+h\) with \(h>0\) moves to the right, outside the interval. A calculation at that point is not a valid secant estimate from the given domain. The correct approach is to use \(b-h\), which lies inside the interval for sufficiently small positive \(h\).
Worked Example: Correcting an Outside-Domain Secant
Let \(f(x)=x^2\) on \([0,2]\). At \(x=2\), an invalid attempt is to use \([f(2+h)-f(2)]/h\). The point \(2+h\) is outside \([0,2]\), so this quotient cannot be used for the endpoint slope of the given function.
Instead choose \(2-h\), where \(0<h<2\). Since \(f(2)=4\), the valid secant slope is
For \(h=0.1\), this gives \(3.9\), and for \(h=0.01\), it gives \(3.99\). The limit as \(h\to0^+\) is \(4\), so the left-hand endpoint slope is \(f'_-(2)=4\). The exact formula \(4-h\) confirms both the numerical estimates and their limit.
How to Analyze an Endpoint Slope
At a left endpoint use points to the right; at a right endpoint use points to the left.
Use the endpoint value and a second point inside the interval, keeping the signed horizontal change in the denominator.
If the quotient approaches a finite number, that number is the one-sided limiting slope.
It gives the slope indicated by secants from within the domain; it does not establish a two-sided derivative without information from the other side.
Why Endpoint Slopes Matter
Intervals often represent the full range of inputs a model permits. For example, time in a process may begin at \(t=0\), or a measured quantity may be restricted to nonnegative values. At the starting or ending input, the graph may still have a meaningful direction from the available side. One-sided secant limits describe that direction without requiring the model to make sense outside its domain.
The essential habit is to check the domain before forming a quotient. At the left endpoint, approach from the right; at the right endpoint, approach from the left. Then preserve the sign of the horizontal change and identify the result as one-sided. These checks prevent an outside-domain calculation from being mistaken for a tangent-slope estimate.
Check Your Understanding
For each question, identify the available side before evaluating or interpreting a quotient.
- For a function defined on \([2,5]\), from which side should secants approach the left endpoint \(2\)?
- Write the secant quotient at the right endpoint \(b\) using \(b-h\), where \(h>0\).
- If the right-hand difference quotient at a left endpoint approaches \(L\), what does \(L\) describe?
- Why does a one-sided endpoint slope not, by itself, determine a two-sided derivative?
- For \(f(x)=x^2\) on \([0,2]\), write a valid secant quotient approaching the right endpoint using a positive distance \(h\).
