Find an equation of the tangent line to the graph of f(x)=x+3/x^2+2
at x=1
A.5x+9y=17
B.5x+3y=9
C.9y-5x=7
D.5x+9y=7
I will assume you meant:
f(x)= (x+3)/(x^2+2)
then f(1) = 4/3, so the point you want is (1, 4/3)
f'(x) = ( (x^2 + 2)(1) - (x+3)(2x) )/(x^2 + 2)^2
f'(1) = (3 - 8)/9 = - 5/9
tangent: y - 4/3 = (-5/9)(x - 1)
simplify to see which of the choices matches
Find an equation of the line tangent to the graph of x^2+(y-x)^3=9 at x=1
To find the equation of the tangent line to the graph of f(x) at x=1, we need to find the slope of the tangent line and a point on the tangent line.
1. Find the derivative of f(x):
f'(x) = (d/dx)(x+3)/(x^2+2)
To find the derivative, we can use the quotient rule. Let's denote u = x+3 and v = x^2+2.
f'(x) = [(v * du/dx) - (u * dv/dx)] / (v^2)
Now, calculate the derivative:
u = x + 3
du/dx = 1
v = x^2 + 2
dv/dx = 2x
f'(x) = [(x^2+2) * 1 - (x+3) * 2x] / [(x^2+2)^2]
= (x^2 + 2 - 2x^2 - 6x) / (x^4 + 4x^2 + 4)
= (-x^2 - 6x + 2) / (x^4 + 4x^2 + 4)
= (-x^2 - 6x + 2) / ((x^2 + 2)(x^2 + 2))
= (-x^2 - 6x + 2) / (x^2 + 2)^2
2. Find the slope of the tangent line:
The slope of the tangent line at x=1 is equal to f'(1). Substitute x=1 into the derivative we found above:
f'(1) = (-1^2 - 6(1) + 2) / (1^2 + 2)^2
= (-1 - 6 + 2) / (1 + 2)^2
= (-5) / (9)
= -5/9
3. Find a point on the tangent line:
To find a point on the tangent line, substitute x=1 into the original function f(x):
f(1) = 1 + 3 / 1^2 + 2
= 4 / 3
So, the point on the tangent line is (1, 4/3).
Now that we have the slope (-5/9) and a point (1, 4/3), we can use the point-slope form of the equation of a line to find the equation of the tangent line.
y - y1 = m(x - x1) where (x1, y1) is the point and m is the slope
Plugging in the values, we get:
y - 4/3 = (-5/9)(x - 1)
Multiplying both sides by 9 to get rid of the fraction, we get:
9y - 12 = -5(x - 1)
Expanding and simplifying, we get:
9y - 12 = -5x + 5
Rearranging the equation to the standard form, we get:
5x + 9y = 17
Therefore, the equation of the tangent line to the graph of f(x) at x=1 is given by option A: 5x + 9y = 17.