Find an equation of the tangent line to the graph at the given point. (The graph is called a Witch of Agnesi.)


f(x)= 7 / x^2 + 3

(2,1)

the slope is -14x/(x^^2+3)^2, so at (2,1) the slope is -28/49 = -4/7

Now you have a point and a slope, so the line is

y-1 = -4/7 (x-2)

See the graphs at

http://www.wolframalpha.com/input/?i=plot+y%3D7%2F(x%5E2%2B3),+y+%3D+-4%2F7+(x-2)%2B1

Well, well, well, it looks like we have a witch on our hands! Let's conjure up the equation of the tangent line at the point (2,1) of the Witch of Agnesi, shall we?

First, we need to find the derivative of f(x) = 7 / (x^2 + 3). So, grab your broomstick and buckle up!

f'(x) = -14x / (x^2 + 3)^2

Now that we have our trusty derivative, let's find the slope of the tangent line at (2,1) by plugging in x = 2 into f'(x):

f'(2) = -14(2) / (2^2 + 3)^2
= -28 / (4 + 3)^2
= -28 / 49
= -4/7

Oh, it seems the slope is a little negative...just like my sense of fashion! (It's all polka dots and oversized shoes, you know?)

Now, to find the equation of the tangent line, we'll use the point-slope formula with the slope (-4/7) and the point (2,1):

y - 1 = (-4/7)(x - 2)

Congratulations! You've successfully summoned the equation of the tangent line to the Witch of Agnesi! Keep those potions brewing and those points plotting!

To find the equation of the tangent line to the graph at the given point (2,1), we need to find the derivative of the function f(x) and then use the derivative to find the slope of the tangent line.

Step 1: Find the derivative of f(x):
To find the derivative, we can use the power rule and the chain rule.
f(x) = 7 / (x^2 + 3)
Using the power rule, the derivative of x^n is n*x^(n-1).
Using the chain rule, the derivative of 1 / (g(x)) is -g'(x) / (g(x))^2.
Therefore, the derivative of f(x) can be found as follows:
f'(x) = -14x / (x^2 + 3)^2

Step 2: Find the slope of the tangent line at the point (2,1):
Substitute x = 2 into the derivative equation:
f'(2) = -14*2 / (2^2 + 3)^2
= -28 / (4 + 3)^2
= -28 / 49

Therefore, the slope of the tangent line at the point (2,1) is -28 / 49.

Step 3: Find the equation of the tangent line:
We have the point (2,1) and the slope (-28 / 49), so we can use the point-slope form to find the equation of the tangent line.
The point-slope form is given by:
y - y1 = m(x - x1)
where (x1, y1) is the point and m is the slope.

Substituting the values, we get:
y - 1 = (-28 / 49)(x - 2)

Simplifying the equation gives us the equation of the tangent line:
y - 1 = (-28/49)x + (56/49)
y = (-28/49)x + (56/49) + 1
y = (-28/49)x + (56/49) + 49/49
y = (-28/49)x + (105/49)

Therefore, the equation of the tangent line to the graph of f(x) = 7 / (x^2 + 3) at the point (2,1) is y = (-28/49)x + (105/49).

To find the equation of the tangent line to the graph at a given point, you will need to follow these steps:

1. Calculate the derivative of the function f(x) with respect to x.
2. Substitute the x-coordinate of the given point into the derivative to find the slope of the tangent line.
3. Use the point-slope form of a linear equation to write the equation of the tangent line.

Let's go through each step to find the equation of the tangent line to the Witch of Agnesi graph at the point (2,1).

Step 1: Calculate the derivative of f(x)
To find the derivative of f(x), you need to use the power rule and the chain rule if necessary.

Given the function f(x) = 7 / (x^2 + 3), we can rewrite it as:
f(x) = 7(x^2 + 3)^(-1)

Now, find the derivative of f(x) using the power rule and chain rule:
f'(x) = -7(x^2 + 3)^(-2) * 2x

Simplifying further:
f'(x) = -14x / (x^2 + 3)^2

Step 2: Substitute the x-coordinate into the derivative
To find the slope of the tangent line at the point (2,1), substitute x = 2 into f'(x):

f'(2) = -14(2) / (2^2 + 3)^2
f'(2) = -28 / 25

So, the slope of the tangent line at the point (2,1) is -28/25.

Step 3: Write the equation of the tangent line using the point-slope form
Now that we know the slope (-28/25) and have a point on the line (2,1), we can use the point-slope form to find the equation of the tangent line.

The point-slope form is:
y - y1 = m(x - x1)

Using the point (2,1) and the slope -28/25, we substitute these values into the equation:
y - 1 = (-28/25)(x - 2)

Expanding and simplifying:
y - 1 = -28/25x + 56/25
y = -28/25x + 56/25 + 1
y = -28/25x + 81/25

Therefore, the equation of the tangent line to the Witch of Agnesi graph at the point (2,1) is y = -28/25x + 81/25.