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Grade 12 (MCV4U) Calculus & Vectors

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The Slope of a Tangent

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Tangent A tangent is the straight line that most resembles the graph near a point. Its slope tells how steep the graph is at the point of tangency. Slope of a Tangent The slope of the tangent to a curve at a point P is the limiting slope of the secant PQ as the point q slides along the curve toward P. In other words, the slope of the tangent is said to be the limit of the slope of the secant as Q approaches P along the curve. Example 1: Slope of a Tangent as a limiting value Find the slope of the tangent to the curve f ( x)  x 2 at the point P when x = 1.

Introduction to Limits y Tangent Line

y Q3

Tangent Line

Q3

Q2 (x, f(x)) Q1

Q2

(a+h, f(a+h)) Q1

p

p y=f(x)

(a, f(a))

y=f(x)

(a, f(a))

x

x a      

x3

x2

As Q  P, Secant PQ is getting closer to bacome Tangent at P xa MPQ  MP f ( x )  f (a ) x a f ( x)  f ( a) M P  lim x a xa

M PQ 

RHHS Mathematics Department

a

x1

h

a+h

Let h be the horizontal displacement between P & Q on x-axis.  As Q  P, 

Secant PQ is getting closer to become Tangent at P

 

h0 MPQ  MP



f ( a  h)  f ( a) h f ( a  h)  f ( a) M P  lim h0 h



M PQ 

Grade 12 (MCV4U) Calculus & Vectors

Page 2 of 2

The Slope of a Tangent

Date:

Slope of a Tangent as a Limit The slope of the tangent to the graph y  f (x) at point P (a, f(a)) is M  lim

 x 0

y f ( a  h)  f (a ) , if this limit exists.  lim  x h 0 h

Example 2: Slope of a Tangent as a limiting value (Cubic Function) Use limits to find the slope of the tangent line to f ( x)  3 x3  2 x  4 at the point when x = -1.

Example 3: Equation of a Tangent as a limiting value (Rational Function) Use limits to find the equation of the tangent line to f ( x)  2x  5 at point (5, 1). x

Example 4: Equation of a Tangent as a limiting value (Radical Function) Use limits to find the equation of the tangent line to f ( x)  x  4 at point x = 8.

RHHS Mathematics Department

Homework: P. 18 #3,6,8b,9c,10b,11,17,19-23...