The equation of the tangent line to f(x) =sqrt{x} at x = 25 can be written in the form y = mx+b
where m =.1
b =?
i don't know how to find the value of b
y = sqrt x = x^(1/2)
dy/dx = (1/2) x^-1/2)
at x = 25
dy/dx = slope = m = (1/2)/5 = 1/10
so
y = .1 x + b
when x = 25, y = 5 so the line goes through (25,5)
5 = .1 (25) = b
5 = 2.5 + b
b = 2.5
so
y = .1 x + 2.5
thank you
Well, finding the value of b is no "sqrt" of a problem! To determine the value of b, we need to substitute the given x-coordinate (x = 25) into the original equation f(x) = √x and calculate the corresponding y-coordinate.
So, let's plug in x = 25:
f(x) = √25
f(x) = 5
Now, we have our point (25, 5) on the curve of f(x) = √x. To find the equation of the tangent line, we need to use the point-slope form, y - y₁ = m(x - x₁), where (x₁, y₁) is our given point (25, 5), and m is the slope.
Given that m = 0.1, let's proceed:
y - 5 = 0.1(x - 25)
Now, we can simplify and rewrite the equation in the y = mx + b form:
y - 5 = 0.1x - 2.5
y = 0.1x + 2.5
Hence, the equation of the tangent line to f(x) = √x at x = 25 can be written as y = 0.1x + 2.5. The value of b is simply 2.5.
I apologize if the math jokes didn't add up, but I hope this explanation helped!
To find the value of b, we need to find the y-coordinate of the point where the tangent line intersects the graph of f(x) = sqrt(x) at x = 25.
Step 1: Find the derivative of f(x) = sqrt(x).
The derivative of f(x) = sqrt(x) can be found using the power rule of differentiation. Let's differentiate f(x):
f'(x) = (1/2) * (1/sqrt(x)) = 1 / (2*sqrt(x))
Step 2: Find the slope of the tangent line.
To find the slope of the tangent line at x = 25, we substitute x = 25 into the derivative:
m = f'(25) = 1 / (2*sqrt(25)) = 1 / (2*5) = 1/10 = 0.1
Step 3: Plug in the slope and the point (25, f(25)) into the point-slope form of a line equation.
Using the point-slope form, we have:
y - f(25) = m(x - 25)
Step 4: Calculate f(25).
To find f(25), substitute x = 25 into the original function f(x):
f(25) = sqrt(25) = 5
Step 5: Substitute the values to get the equation in the form y = mx + b.
y - 5 = 0.1(x - 25)
Step 6: Simplify the equation to solve for b.
Distribute 0.1 into (x - 25):
y - 5 = 0.1x - 0.1*25
y - 5 = 0.1x - 2.5
Rearrange the equation to solve for y:
y = 0.1x - 2.5 + 5
y = 0.1x + 2.5
The equation of the tangent line to f(x) = sqrt(x) at x = 25 can be written in the form y = mx + b, where m = 0.1 and b = 2.5.
To find the value of b, we need to determine the y-coordinate of the point where the tangent line intersects the curve of the function f(x) = sqrt(x) at x = 25.
First, let's find the slope of the tangent line, m. Given that m = 0.1, we know that the slope of the tangent line is 0.1.
Now, let's find the y-coordinate of the point on the curve at x = 25. We can do this by substituting x = 25 into the function f(x) = sqrt(x):
f(25) = sqrt(25) = 5
So, the point on the curve is (25, 5), where x = 25 and y = 5.
Now we can use the point-slope form of a linear equation, which is given by:
y - y1 = m(x - x1)
Using our values, we have:
y - 5 = 0.1(x - 25)
Now let's simplify the equation:
y - 5 = 0.1x - 2.5
Next, let's isolate y:
y = 0.1x - 2.5 + 5
y = 0.1x + 2.5
Comparing this equation with the form y = mx + b, we can see that b = 2.5.
Therefore, the equation of the tangent line to f(x) = sqrt(x) at x = 25 can be written as y = 0.1x + 2.5, where m = 0.1 and b = 2.5.