Find the slope m and an equation of the tangent line to the graph of the function f at the specified point.
f(x)=x^2/x+4 ;(1,1/5)
m =??
y =??
Oh, slope me silly! It's time to find the slope and equation of the tangent line. Let's get cracking, shall we?
To find the slope, we can use differentiation. Taking the derivative of f(x) = x^2/(x+4) will give us the slope.
So, using some math magic, we find that the derivative of f(x) is: (2x+8)/(x+4)^2.
Now, let's substitute x = 1 into the derivative to get the slope at the point (1, 1/5). Plugging it in, we have:
m = (2(1) + 8)/(1+4)^2.
m = (2 + 8)/(5)^2.
m = 10/25.
m = 2/5.
The slope, my friend, is 2/5. Isn't math amusing?
Now, for the equation of the tangent line. We'll use the point-slope form of a line, y - y1 = m(x - x1), where (x1, y1) is the given point.
Plugging in the values (1, 1/5) and m = 2/5, we have:
y - 1/5 = 2/5(x - 1).
Let's distribute the 2/5 to get rid of those pesky fractions:
y - 1/5 = 2/5x - 2/5.
Then, let's add 1/5 to both sides to get y alone:
y = 2/5x - 2/5 + 1/5.
Simplifying that, we have:
y = 2/5x - 1/5.
Voila! There's your equation of the tangent line, my dear friend. Hope you had a laugh along the way!
To find the slope, m, of the tangent line to the graph of the function f at the point (1, 1/5), we need to find the derivative of f(x) first.
The derivative of f(x) can be found using the quotient rule. The quotient rule states that if we have a function in the form f(x) = g(x)/h(x), then the derivative of f(x) is given by:
f'(x) = (g'(x) * h(x) - g(x) * h'(x)) / (h(x))^2
For the function f(x) = x^2/(x+4), we can rewrite it as f(x) = x^2 * (x+4)^-1.
Applying the quotient rule, we can find the derivative of f(x):
f'(x) = (2x * (x+4)^-1) + (x^2 * (-1) * (x+4)^-2)
Simplifying this expression, we get:
f'(x) = (2x) / (x+4) - (x^2) / (x+4)^2
Now, substitute x = 1 into the derivative expression to find the slope, m, of the tangent line at the point x = 1:
m = f'(1) = (2(1)) / (1+4) - (1^2) / (1+4)^2
m = 2/5 - 1/25
m = 50/125 - 5/125
m = 45/125
m = 9/25
Therefore, the slope m of the tangent line to the graph of f at the point (1, 1/5) is 9/25.
To find the equation of the tangent line using the point-slope form, we can use the coordinates of the point (1, 1/5) and the slope m we found:
y - y1 = m(x - x1)
where x1 = 1 and y1 = 1/5.
Substituting the values, we get:
y - (1/5) = (9/25)(x - 1)
Simplifying, we get:
y - 1/5 = 9/25x - 9/25
Moving the constant term to the right side:
y = 9/25x - 9/25 + 1/5
y = 9/25x - 9/25 + 5/25
y = 9/25x - 4/25
Therefore, the equation of the tangent line to the graph of f at the point (1, 1/5) is y = 9/25x - 4/25.
To find the slope, m, of the tangent line to the graph of the function f at a given point, you can use the derivative of the function. The derivative represents the rate of change of the function at any point and provides the slope of the tangent line.
Let's find the derivative of f(x) = x^2 / (x + 4) first. We can use the quotient rule for differentiation.
f(x) = x^2 / (x + 4)
Using the quotient rule: (f(x)g'(x) - g(x)f'(x)) / (g(x))^2
where f(x) = x^2 and g(x) = (x + 4),
f'(x) = (2x)*(x + 4) - (x^2)*(1) / (x + 4)^2
= (2x^2 + 8x - x^2) / (x + 4)^2
= (x^2 + 8x) / (x + 4)^2
The derivative of f(x) is (x^2 + 8x) / (x + 4)^2.
Now, let's find the value of the slope, m, at the specified point (1, 1/5). To do this, substitute x = 1 into the derivative:
m = ((1)^2 + 8(1)) / ((1) + 4)^2
= (1 + 8) / (5)^2
= 9/25
Therefore, the slope, m, is 9/25.
To find the equation of the tangent line, we can use the point-slope form: y - y1 = m(x - x1), where (x1, y1) is the given point.
In this case, the given point is (1, 1/5) and the slope, m, is 9/25. Plugging these values into the formula:
y - (1/5) = (9/25)(x - 1)
Expanding and simplifying:
y - 1/5 = (9/25)x - 9/25
y = (9/25)x - 9/25 + 1/5
y = (9/25)x - 9/25 + 5/25
y = (9/25)x - 4/25
Therefore, the equation of the tangent line to the graph of the function f at the point (1, 1/5) is y = (9/25)x - 4/25.
dy/dx = slope = [ (x+4)(2x) - x^2 ] /(x+4)^2
= ( x^2 + 8 x ] / (x+4)^2
at x = 1 this slope is
= 9/25 so that is m
goes through (1, 1/5)
1/5 = (9/25)(1) + b
5/25 = 9/25 + b
b = -4/25
so
y = 9 x/25 - 4/25
or
25 y = 9 x - 4