{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

2.1 The Derivative and the Tangent Line Problem

# 2.1 The Derivative and the Tangent Line Problem - The...

This preview shows pages 1–10. Sign up to view the full content.

The Derivative The Derivative and the Tangent and the Tangent Line Problem Line Problem Section 2-1

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Slope of Secant Line Slope of Secant Line x x ) ( ) ( c f x c f - + (c, f(c)) (c + ,f(c + ) x x x c f x c f m - + = ) ( ) ( sec
Definition of Tangent Line with Definition of Tangent Line with Slope Slope m m If f is defined on an open interval containing c, and if the limit exists, then the line passing through (c, f(c)) with slope m is the tangent line to the graph of f at the point (c, f(c)). m x c f x c f x y x x = - + = ) ( ) ( lim lim 0 0

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Vertical Tangent Lines Vertical Tangent Lines The defn. of a tangent line to a curve does not cover the possibility of a vertical tangent line. If f is continuous at c and The vertical line x = c, passing through (c, f(c)) is a vertical tangent line to the graph of f. - = - + or x c f x c f x ) ( ) ( lim 0
Definition of the Derivative of Definition of the Derivative of a Function a Function

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: The derivative of f at x is given by provided the limit exists. x x f x x f x f x ∆-∆ + = → ∆ ) ( ) ( lim ) ( ' Notation for Derivatives Notation for Derivatives ) ( ' x f dx dy ' y [ ] ) ( x f dx d [ ] y D x Alternative form of the Alternative form of the Derivative Derivative c x c f x f c f c x--= → ) ( ) ( lim ) ( ' Thm. 2.1 Differentiability Thm. 2.1 Differentiability Implies Continuity Implies Continuity If f is differentiable at x = c, then f is continuous at x = c. It is possible for a function to be continuous at x = c and not be differentiable at x = c. Thus, continuity does not imply differentiability. Joke Time Joke Time What would America be called if we all drove pink cars? A pink carnation! Why should you never play cards in the jungle? It’s full of cheetahs!...
View Full Document

{[ snackBarMessage ]}

### Page1 / 10

2.1 The Derivative and the Tangent Line Problem - The...

This preview shows document pages 1 - 10. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online