From Secant Lines to a Tangent
A secant line passes through two points on a curve. If one point stays fixed while the other moves along the curve toward it, the secant lines may approach a tangent line. The slope of that tangent is the limit of the secant slopes, when that limit exists and is finite.
This idea gives a precise meaning to tangent slope at a point. It also explains why a tangent slope describes the graph locally: it records the direction toward which the nearby secants settle.
The numerator is the change in vertical coordinate, and the denominator is the change in horizontal coordinate. Their ratio is the slope between the two graph points. The moving point must have \(x\ne a\), because if \(x=a\), the two points coincide and do not determine a secant line.
Why a Limit Gives the Tangent Slope
Imagine keeping \(P\) fixed and moving \(Q\) closer to it along the curve. Each position of \(Q\) gives a different secant. If the secant slopes approach \(L\), their directions approach the direction of a line with slope \(L\). That limiting direction is the tangent direction.
The line equation makes the connection precise. Use \(t\) for a variable horizontal coordinate on each secant line. The secant through \(P\) has equation \(y-f(a)=m_{\mathrm{sec}}(x)(t-a)\). If \(m_{\mathrm{sec}}(x)\) approaches \(L\), then for each fixed \(t\), the right-hand side approaches \(L(t-a)\). The limiting line therefore has equation \(y-f(a)=L(t-a)\): it passes through \(P\) and has slope \(L\). This explains why the limit of the secant slopes determines the tangent slope.
Worked Example: Secants on the Graph of a Parabola
Worked Example: Secants on the Graph of a Parabola
Consider \(f(x)=x^2\) at \(a=1\). Find several secant slopes and use them to identify the limiting tangent slope.
The fixed point is \(P=(1,f(1))=(1,1)\). For any \(x\ne1\), the moving point is \(Q=(x,x^2)\), so
The cancellation is valid because \(x\ne1\). Now calculate secant slopes for inputs on both sides of \(1\):
- At \(x=2\), \(Q=(2,4)\), so \(m_{\mathrm{sec}}=\frac{4-1}{2-1}=\frac{3}{1}=3\).
- At \(x=1.5\), \(Q=(1.5,2.25)\), so \(m_{\mathrm{sec}}=\frac{2.25-1}{1.5-1}=\frac{1.25}{0.5}=2.5\).
- At \(x=1.1\), \(Q=(1.1,1.21)\), so \(m_{\mathrm{sec}}=\frac{1.21-1}{1.1-1}=\frac{0.21}{0.1}=2.1\).
- At \(x=0.9\), \(Q=(0.9,0.81)\), so \(m_{\mathrm{sec}}=\frac{0.81-1}{0.9-1}=\frac{-0.19}{-0.1}=1.9\).
- At \(x=0.5\), \(Q=(0.5,0.25)\), so \(m_{\mathrm{sec}}=\frac{0.25-1}{0.5-1}=\frac{-0.75}{-0.5}=1.5\).
The slopes from the right decrease toward \(2\), while those from the left increase toward \(2\). The exact expression \(x+1\) confirms the limit: as \(x\) approaches \(1\), \(x+1\) approaches \(2\). Thus the tangent slope at \((1,1)\) is \(2\).
To check, the line through \((1,1)\) with slope \(2\) is \(y-1=2(x-1)\), or \(y=2x-1\). It passes through the point because \(2(1)-1=1\), and its slope is \(2\), as required. Its contact with the parabola is also consistent with tangency: \(x^2-(2x-1)=(x-1)^2\), which is zero at \(x=1\).
A graph or table of nearby slopes can make the trend visible, but finitely many values alone cannot establish a limit. In this example, the formula \(m_{\mathrm{sec}}(x)=x+1\) verifies that the trend continues for every \(x\ne1\) near \(1\). The graph-based interpretation and the algebraic limit support the same conclusion.
Worked Example: A Straight Line Has the Same Secant Slope Everywhere
Worked Example: A Straight Line Has the Same Secant Slope Everywhere
Let \(f(x)=3x+2\), and find the tangent slope at \(a=1\) from the secants.
The fixed point is \(P=(1,f(1))=(1,5)\). Choose \(x=2\). Then \(Q=(2,8)\), and the secant slope is \(\frac{8-5}{2-1}=\frac{3}{1}=3\).
Choose a point from the other side, \(x=0\). Then \(Q=(0,2)\), and the secant slope is \(\frac{2-5}{0-1}=\frac{-3}{-1}=3\). More generally, for every \(x\ne1\),
Every secant already has slope \(3\), so its limit as \(x\) approaches \(1\) is \(3\). The tangent line is \(y-5=3(x-1)\), which simplifies to \(y=3x+2\). Substituting \(x=1\) gives \(y=5\), and the line’s slope is \(3\). Thus the tangent agrees with the original straight line, as expected.
When the Two Sides Do Not Agree
The limit must describe what happens as the moving input approaches \(a\) from both sides. A graph with a sharp corner can have different limiting secant slopes on the left and right, so it has no single tangent slope there.
Worked Example: A Corner with No Single Tangent Slope
Consider \(f(x)=|x|\) at \(a=0\). The fixed point is \(P=(0,0)\). For \(x>0\), \(f(x)=x\), so the secant slope is \(\frac{|x|-|0|}{x-0}=\frac{x}{x}=1\).
For \(x<0\), \(f(x)=-x\), so the secant slope is \(\frac{|x|-|0|}{x-0}=\frac{-x}{x}=-1\). For instance, at \(x=0.2\), the secant slope is \(\frac{0.2}{0.2}=1\); at \(x=-0.2\), it is \(\frac{0.2}{-0.2}=-1\).
As \(x\) approaches \(0\) from the right, the slopes remain \(1\). From the left, they remain \(-1\). Since \(1\ne-1\), the two-sided limit does not exist. The graph has no single tangent slope at the corner. Each one-sided line through the vertex has the correct slope for that side, but neither supplies a common two-sided tangent.
Check the point involved: both side lines pass through \((0,0)\). The failure is not a misplaced point; it is the disagreement between their slopes.
A Common Mistake: Substituting the Fixed Input Too Soon
In the secant-slope expression, putting \(x=a\) makes both graph points the same. The resulting fraction has zero in its denominator and does not represent a secant slope. It is a mistake to interpret that substitution as the tangent slope.
The order matters: first calculate secant slopes for distinct points, then determine their limiting value as the moving point approaches the fixed point. A limit is about nearby values; it does not require evaluating the original quotient at the point where it is undefined.
What the Limit Tells You
A finite limit of secant slopes gives the tangent’s direction and slope at the fixed point. If slopes from both sides converge to the same number, that number is the tangent slope. If they approach different numbers, as at the corner of \(y=|x|\), no single tangent slope exists under this definition.
The central reasoning is geometric but precise: two points determine a secant; moving one point toward the other makes the secant direction change; and when those slopes settle toward one finite value, that value is the slope of the tangent. The limit is what turns the visual idea of “touching the curve” into a test based on nearby graph points.
Check Your Understanding
Use the fixed-point and moving-point idea in each question. For calculations, show the coordinates or the secant-slope expression you use.
- For \(f(x)=x^2\) at \(a=2\), find the secant slope when \(x=2.1\).
- For the same function and point, find the secant slope when \(x=1.9\). What does the pair of slopes suggest about the tangent slope?
- Why is substituting \(x=a\) directly into the secant-slope quotient not a valid way to find a tangent slope?
- For \(f(x)=|x|\) at \(a=0\), what are the slopes approached from the right and from the left?
- State the condition on the one-sided secant-slope limits that allows a finite tangent slope at an interior point.
