Answered: find (a) the slopeof the curve at the…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Question

find (a) the slope
of the curve at the given point P, and (b) an equation of the tangent
line at P. y = x3, P(2, 8)

Expert Solution

Step 1

Concept:

A definition of a function is that a relation from a set of inputs to a set of possible outputs where each inputs is related to exactly one output. A function assigns only one output for each one.

Step 2

Given:

$y = x^{3} a n d P (2, 8)$

Step 3

$P r i n c i p l e i d e a s : L e t f (x) b e a f u n c t i o n, a n d l e t P b e a p o i n t w i t h c o o r d i n a t e s (x, f (x)) . A m e t h o d t o f i n d t h e s l o p e, o r " i n s tantan e o u s r a t e o f c h a n g e ", o f f a t p o i s a s f o l l o w s : c o n s i d e r i n g a n o t h e r p o i n t Q, w i t h c o o r d i n a t e s (x + h, f (x + h)) w h e r e, h i s a r e a l n u m b e r . I t i s c o m p u t e d t h a t t h e s e c e n t s l o p e b e t w e e n P a n d Q a s \frac{∆ y}{∆ x} = \frac{f (x + h) - f (x)}{h} . A s Q a p p r o a c h e s P, h a p p r o a c h e s 0, a n d t h e s e c a n t s l o p e w i l l a p p r o a c h t h e \tan g e n t s l o p e a t p o i n t P . A l s o, g i v e n a p o i n t (a, b), t h e l i n e c o n t a i n i n g t h a t p o i n t a n d h a v i n g s l o p e m w i l l h a v e t h e e q u a t i o n y = b + m (x - a) . T h i s f o r m i s m o s t c o m m o n l y r e f e r r e d t o a s t h e " p o i n t s l o p e e q u a t i o n " .$

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