The equation of the tangent line to f(x) = \sqrt{x} at x = 64 is y =
f(64) = 8
f'(64) = 1/16
so, the line is
y-8 = 1/16 (x-64)
To find the equation of the tangent line to the function f(x) = √x at x = 64, we can use the slope-intercept form of a line, which is y = mx + b, where m is the slope of the line and b is the y-intercept.
Step 1: Find the slope of the tangent line
The slope of the tangent line can be found using the derivative of the function. Taking the derivative of f(x) = √x, we get:
f'(x) = (1/2) * (x^(-1/2))
Evaluate the derivative at x = 64:
f'(64) = (1/2) * (64^(-1/2))
Simplify the expression:
f'(64) = (1/2) * (1/8)
f'(64) = 1/16
So, the slope of the tangent line at x = 64 is 1/16.
Step 2: Find the y-coordinate of the point on the function at x = 64
To find the y-coordinate, substitute x = 64 into the function:
f(64) = √64
Simplify the expression:
f(64) = 8
So, the point on the function f(x) = √x at x = 64 is (64, 8).
Step 3: Write the equation of the tangent line
Now that we have the slope (m = 1/16) and a point on the line (64, 8), we can use the point-slope form of the line to find the equation:
y - y1 = m(x - x1)
Substitute the values:
y - 8 = (1/16)(x - 64)
Simplify the equation:
y - 8 = (1/16)x - 4
Rearrange the equation:
y = (1/16)x + 4
So, the equation of the tangent line to f(x) = √x at x = 64 is y = (1/16)x + 4.
To find the equation of the tangent line to the function f(x) = √x at the point x = 64, we can follow these steps:
Step 1: Find the derivative of the function f(x). The derivative of √x can be found using the power rule of differentiation.
As a reminder, the power rule states that the derivative of x^n is n*x^(n-1), where n is a constant.
In this case, we have f(x) = x^(1/2), so applying the power rule, we get:
f'(x) = (1/2) * x^(-1/2)
Step 2: Evaluate the derivative at x = 64. Plug in x = 64 into f'(x):
f'(64) = (1/2) * 64^(-1/2)
= (1/2) * 1/(√64)
= (1/2) * 1/8
= 1/16
So the slope of the tangent line to f(x) at x = 64 is 1/16.
Step 3: Use the point-slope form of a linear equation to find the equation of the tangent line. The point-slope form is given by:
y - y1 = m(x - x1)
where (x1, y1) is a point on the line and m is the slope of the line.
We have the slope, m = 1/16, and the point (x1, y1) = (64, f(64)) on the line.
Plugging these values into the point-slope form, we get:
y - f(64) = (1/16)(x - 64)
Now, to simplify the equation, we need to find f(64). Plugging x = 64 into the original function f(x) = √x:
f(64) = √64
= 8
Therefore, the equation of the tangent line to f(x) = √x at x = 64 is:
y - 8 = (1/16)(x - 64)