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 = 2 - x3, P(1, 1)

Expert Solution

Step 1

We have to find:

$(a)$ slope of the curve at the given point $P$

and $(b)$ an equation of tangent at the given point $P$

Where curve and point is given

$y = 2 - x^{3}, P (1, 1)$

$(a)$ We know that slope of the curve and derivative of the curve at the given point is same

or,

$m = {\frac{d y}{d x}}_{(a t p o i n t P)}$

Therefore differentiating the curve with respect to $' x'$ , we get

$\begin{array}{rcl} \frac{d y}{d x} & = & \frac{d (2 - x^{3})}{d x} \\ = & \frac{d 2}{d x} - \frac{d x^{3}}{d x} \\ = & 0 - 3 x^{3 - 1} \\ = & - 3 x^{2} \end{array}$

Finding $\frac{d y}{d x}$ at point $P (1, 1)$ ,

$\begin{array}{rcl} \frac{d y}{d x} & = & - 3 x^{2} \\ = & - 3 {(1)}^{2} \\ = & - 3 \end{array}$

Hence, slope of the curve is $- 3$ .

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