Let f (x) = Vx.Find the slope m and the point-slope equation of the line tangent to the graph of f at the point (49, 7)m =limlimX--definitionsimplifiedexact valueEquation of tangent line:y =x +*: symbolic formatting help

Consider the given function fx=x Suppose m be the slope of the line tangent to the graph of the…

Answered: Let f (x) = Vx. Find the slope m and…

Let f (x) = Vx. Find the slope m and the point-slope equation of the line tangent to the graph of f at the point (49, 7)

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter3: Functions And Graphs

Section3.3: Lines

Problem 41E

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Concept explainers

Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

Slope

The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:

Question

Let f (x) = Vx.
Find the slope m and the point-slope equation of the line tangent to the graph of f at the point (49, 7)
m =
lim
lim
X--
definition
simplified
exact value
Equation of tangent line:
y =
x +
*: symbolic formatting help

Expert Solution

Step 1

Consider the given function

$f (x) = \sqrt{x}$

Suppose m be the slope of the line tangent to the graph of the function $f (x)$

Then, the slope m of the line tangent to the graph of the function $f (x)$ at $x = c$ is,

$m = \lim_{x \to c} \frac{f (x) - f (c)}{x - c}$

Now,

$\begin{array}{rcl} m & = & \lim_{x \to 49} \frac{f (x) - f (49)}{x - 49} \\ = & \lim_{x \to 49} \frac{\sqrt{x} - \sqrt{49}}{x - 49} \\ = & \lim_{x \to 49} \frac{\sqrt{x} - 7}{x - 49} \\ = & \lim_{x \to 49} \frac{\sqrt{x} - 7}{(\sqrt{x} + 7) (\sqrt{x} - 7)} (S i n c e, (a + b) (a - b) = a^{2} - b^{2}) \\ = & \lim_{x \to 49} \frac{1}{(\sqrt{x} + 7)} \\ = & \frac{1}{(\sqrt{49} + 7)} \\ = & \frac{1}{(7 + 7)} \\ = & \frac{1}{14} \end{array}$

Hence, the required slope of the line tangent to the graph of the function $f (x)$ at the point $(49, 7)$ is $\begin{array}{rcl} m & = & \lim_{x \to c} \frac{f (x) - f (c)}{x - c} = \lim_{x \to 49} \frac{1}{(\sqrt{x} + 7)} = \frac{1}{14} \end{array}$

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