Understanding Tangent Lines in Mathematics

1. Tangent Lines(Sections 1.4 and 2.1 ) Alex Karassev

2. Tangent line • What is a tangent line to a curve on the plane? • Simple case: for a circle, a line that has only one common point with the circle is called tangent line to the circle • This does not work in general! ? P P

3. Idea: approximate tangent line by secant lines • Secant line intersects the curve at the point P and some other point, Px y y Px Px P P x x a x x a

4. Tangent line as the limit of secant lines • Suppose the first coordinate of the point P is a • As x → a, Px x → a, and the secant line approaches a limiting position, which we will call the tangent line y y Px Px P P x x a x x a

5. Slope of the tangent line • Since the tangent line is the limit of secant lines, slope of the tangent line is the limit of slopes of secant lines • P has coordinates (a,f(a)) • Px has coordinates (x,f(x)) • Secant line is the linethrough P and Px • Thus the slopeof secant line is: m y y=f(x) mx P Px f(x) x a x

6. Slope of the tangent line • We define slope m of the tangent line as the limit of slopes of secant lines as x approaches a: • Thus we have: m y y=f(x) mx P Px f(x) x a x

7. Example • Find equation of the tangent line to curve y=x2 at the point (2,4) y P Px x 2

8. Solution • We already know that the point (2,4) is on the tangent line, so we need to find the slope of the tangent line • P has coordinates (2,4) • Px has coordinates (x,x2) • Thus the slope of secant line is: y P Px x 2

9. Solution • Now we compute the slope of the tangent line by computing the limit as x approaches 2: y P Px x 2

10. Solution • Thus the slope of tangent line is 4 and therefore the equation of the tangent line is y – 4 = 4 (x – 2) , or equivalently y = 4x – 4 y P Px x 2