Download presentation

Presentation is loading. Please wait.

Published byMohamed Hood Modified over 2 years ago

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

Similar presentations

OK

Aim: What do slope, tangent and the derivative have to do with each other? Do Now: What is the equation of the line tangent to the circle at point (7,

Aim: What do slope, tangent and the derivative have to do with each other? Do Now: What is the equation of the line tangent to the circle at point (7,

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on consumer protection act 1986 india Ppt on bluetooth architecture piconets Ppt on mauryan administration Ppt on art and craft movement wikipedia Ppt on social etiquettes Ppt on 15th august independence day Ppt on new technology in mechanical Ppt on phonetic transcription of english words Ppt on inhabiting other planets with water Ppt on law against child marriage laws