Consider the following function.
f(x)= x^2 - sqrt(x)
Find the equation of the line tangent at the point (1, 0).
the answer would be 6, duh
To find the equation of the line tangent to the function at the point (1, 0), we need to use the concept of the derivative. The derivative represents the slope of the function at any given point.
To start, let's find the derivative of the function f(x) = x^2 - sqrt(x).
Step 1: Find the derivative of x^2.
The derivative of x^2 with respect to x is 2x.
Step 2: Find the derivative of sqrt(x).
The derivative of sqrt(x) with respect to x is 1/(2*sqrt(x)).
Step 3: Combine the derivatives.
Since our original function has both terms, we subtract the derivative of the second term from the derivative of the first term:
f'(x) = 2x - (1/(2*sqrt(x)))
To find the slope of the tangent line at the point (1, 0), substitute x = 1 into the derivative:
f'(1) = 2(1) - (1/(2*sqrt(1)))
f'(1) = 2 - (1/2)
f'(1) = 1.5
So, the slope of the tangent line at the point (1, 0) is 1.5.
Now, we have the point (1, 0) and the slope 1.5. To find the equation of the line, we can use the point-slope form:
y - y1 = m(x - x1),
where (x1, y1) is the given point and m is the slope.
Plugging in the values:
y - 0 = 1.5(x - 1)
Simplifying:
y = 1.5x - 1.5
Therefore, the equation of the tangent line at the point (1, 0) is y = 1.5x - 1.5.