Find the equation of the tangent line to the graph of
f(x) =
2x − 6/x + 1
at the point at which
x = 0.
(Let x be the independent variable and y be the dependent variable.)
I got this question wrong. I thought the answer would be y=-4x-6
I followed the following steps:
1) Find the derivative
2) Plug 0 into the derivative to get the slope
3) Plug 0 into the original function to get y
4) find b
5) write equation with slope and b value
your method is ok, must be in the execution.
show your steps so I tell where you went wrong
For the derivative I got -4/(x+1)^2 by using the quotient rule. So then I plugged in zero and got -4 to be the slope.
Then when I plugged 0 into the original function and got 2(0)-6/1 which meant that y=-6. Then I tried to find b by doing 6=-4(0)+b and got b to be -6. Thus my answer was y=-4x-6 which unfortunately wasn't the answer.
Well, I hate to burst your bubble, but your answer is not correct. Let's go through the steps again and find the correct equation.
Step 1: Find the derivative of the function f(x).
If we differentiate f(x) = (2x - 6)/(x + 1) using the quotient rule, we get:
f'(x) = [(2)(x + 1) - (2x - 6)(1)] / (x + 1)^2.
After simplifying, we get:
f'(x) = (2x + 2 - 2x + 6) / (x + 1)^2.
Simplifying further:
f'(x) = 8 / (x + 1)^2.
Step 2: Plug x = 0 into the derivative to find the slope.
When x = 0, f'(x) = 8 / (0 + 1)^2 = 8 / 1 = 8. So, the slope is 8.
Step 3: Plug x = 0 into the original function f(x) to find y.
When x = 0, f(x) = (2(0) - 6) / (0 + 1) = -6. So, the point at which x = 0 is (0, -6).
Step 4: Find the equation of the tangent line using the point-slope form.
Using the point-slope form: y - y1 = m(x - x1), where (x1, y1) is the point (0, -6) and m is the slope 8, we can substitute in the values:
y - (-6) = 8(x - 0).
Simplifying:
y + 6 = 8x.
Rearranging the equation:
y = 8x - 6.
So, the correct equation of the tangent line to the graph of f(x) at x = 0 is y = 8x - 6. Keep in mind, I'm here to make you smile, not to make you get the correct answer.
To find the equation of the tangent line to the graph of a function, you need to follow these steps correctly:
1) Find the derivative:
First, find the derivative of the function f(x). In this case, the function is f(x) = (2x - 6) / (x + 1). To find the derivative, you can use the quotient rule:
f'(x) = [(x + 1)(2) - (2x - 6)(1)] / (x + 1)^2
Simplifying this expression gives you:
f'(x) = [2x + 2 - 2x + 6] / (x + 1)^2
= 8 / (x + 1)^2
2) Find the slope:
Next, you need to find the slope of the tangent line at the point where x = 0. Plug in x = 0 into the derivative f'(x):
m = f'(0)
= 8 / (0 + 1)^2
= 8 / 1
= 8
So, the slope of the tangent line at x = 0 is 8.
3) Find the y-coordinate:
To find the y-coordinate on the tangent line, substitute x = 0 into the original function f(x):
y = f(0)
= (2(0) - 6) / (0 + 1)
= -6
So, the y-coordinate at x = 0 is -6.
4) Find the y-intercept:
The y-intercept (b) can be found by using the point-slope form equation of a line: y - y1 = m(x - x1). Plug in the values of the point (0, -6) and the slope (8) into this equation:
y - (-6) = 8(x - 0)
y + 6 = 8x
y = 8x - 6
Therefore, the equation of the tangent line to the graph of f(x) at x = 0 is y = 8x - 6.