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The graph of the function

f(x)=\frac{1}{1+x^{2}}

is called a witch of Maria Ag...

Jan 31, 2024

The graph of the function

f(x)=\frac{1}{1+x^{2}}

is called a witch of Maria Agnesi. (a) Find an equation of the tangent line to the graph of this function at the point

\left(1, \frac{1}{2}\right)

y=

(b) Illustrate your solution to part (a) by graphing the function and the tangent line in the same viewing window.

Generated Graph

Solution by Steps

step 1

To find the equation of the tangent line, we first need the derivative of the function at the point of tangency

step 2

The derivative of

f(x) = \frac{1}{1+x^2}

f'(x) = -\frac{2x}{(1+x^2)^2}

as per the asksia-ll calculation

step 3

Evaluate the derivative at

x = 1

to find the slope of the tangent line:

f'(1) = -\frac{2(1)}{(1+1^2)^2} = -\frac{2}{4} = -\frac{1}{2}

step 4

Use the point-slope form of a line with point

(1, \frac{1}{2})

and slope

-\frac{1}{2}

y - \frac{1}{2} = -\frac{1}{2}(x - 1)

step 5

Simplify the equation of the tangent line to get

y = -\frac{1}{2}x + 1

Answer

y = -\frac{1}{2}x + 1

Key Concept

Finding the equation of a tangent line

Explanation

The equation of the tangent line to a curve at a given point is found using the derivative of the function at that point to determine the slope, and then applying the point-slope form of a line.

Solution by Steps

step 6

To illustrate the solution, plot the function

f(x) = \frac{1}{1+x^2}

and the tangent line

y = -\frac{1}{2}x + 1

on the same graph

Answer

The graph should show the curve

f(x) = \frac{1}{1+x^2}

with the tangent line

y = -\frac{1}{2}x + 1

intersecting it at the point

(1, \frac{1}{2})

Key Concept

Graphing a function and its tangent line

Explanation

The graph provides a visual representation of the function and its tangent line, showing the point of tangency and the slope of the tangent line at that point.