Derivatives
From secant to tangent
Change the interval. Observe the slope. What do you think happens as the two points get closer?
A secant joins two points on a curve. Bring them closer to discover how its slope approaches the tangent slope. We will explore f(x) = x², first at a = 1 and then at a = −2.
1. Two points, one secant
1 / 4f(x)=x^2 Fix P at (1, 1) and move Q along the parabola with a positive increment h. The secant slope is the change in height divided by the change in x. As h gets smaller, the slope approaches 2.
\frac{f(a+h)-f(a)}{h}=2a+h,\quad h\ne0 - Starting point a
- a = 1
- Secant slope
- 3
- Tangent slope
- 2
P is fixed at (a, a²); Q moves along the parabola. The solid secant joins P and Q. The dashed line is the tangent at P. At h = 0, only the tangent remains.
2. From the other side
2 / 4f(x)=x^2 Now Q approaches P from the left, so h is negative. The secant slope is 2 + h: it approaches 2 from below. Compare this with the positive increments in the previous slide.
\frac{f(a+h)-f(a)}{h}=2a+h,\quad h\ne0 - Starting point a
- a = 1
- Secant slope
- 1
- Tangent slope
- 2
P is fixed at (a, a²); Q moves along the parabola. The solid secant joins P and Q. The dashed line is the tangent at P. At h = 0, only the tangent remains.
3. The tangent limit
3 / 4f(x)=x^2 Move h all the way to zero. P and Q coincide, so the secant quotient is undefined. The dashed tangent remains: its slope is the limit of the secant slopes, not the value of the quotient at zero.
\frac{f(a+h)-f(a)}{h}=2a+h,\quad h\ne0 - Starting point a
- a = 1
- Secant slope
- 3
- Tangent slope
- 2
P is fixed at (a, a²); Q moves along the parabola. The solid secant joins P and Q. The dashed line is the tangent at P. At h = 0, only the tangent remains.
4. A different starting point
4 / 4f(x)=x^2 Move P to (−2, 4). For any starting point a on this parabola, the secant slope is 2a + h and the tangent slope is 2a. Here the tangent slopes downward.
\frac{f(a+h)-f(a)}{h}=2a+h,\quad h\ne0 - Starting point a
- a = -2
- Secant slope
- -3
- Tangent slope
- -4
P is fixed at (a, a²); Q moves along the parabola. The solid secant joins P and Q. The dashed line is the tangent at P. At h = 0, only the tangent remains.