Tangent Lines (Sections 2.1 and 3.1 ) Alex Karassev

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

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

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

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 line through P and Px Thus the slope of secant line is: m y y=f(x) mx P Px f(x) x x a

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 x a

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

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

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

Solution y 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 P Px x 2