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

2. P 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 ?

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

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

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)) P x has coordinates (x,f(x)) Secant line is the line through P and Px Thus the slope of secant line is: x y a P PxPx x f(x) y=f(x) mxmx m

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: x y a P PxPx x f(x) y=f(x) mxmx m

7. Example Find equation of the tangent line to curve y=x 2 at the point (2,4) x y P PxPx 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 x y P PxPx 2 P has coordinates (2,4) P x has coordinates (x,x 2 ) Thus the slope of secant line is:

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