Find the equation of the tangent line to the curve y=sqrt(x) at x = c. Put the answer in slope intercept form y=mx+b. Use the definition of the slope of the tangent with no short cuts.
so the point of contact is (c, √c)
y = √x = x^(1/2)
dy/dx = (1/2)x^(-1/2)
at our point
dy/dx = (1/2)c^(-1/2) = 1/(2√c) = slope = m
y = (1/(2√c) x + b
at(c, √c)
√c = (1/(2√c)(c) + b
times 2√c
2c = c + 2√c b
c = 2√c b
b = c/√c
y = (1/(2√c)) x + c/√c
check my arithmetic
You could rationalize the denominator if you had to.
To find the equation of the tangent line to the curve y = sqrt(x) at x = c, we need to first find the slope of the tangent at that point.
Step 1: Find the derivative of the function y = sqrt(x).
The derivative of sqrt(x) can be found by using the power rule. Here, the power is 1/2:
dy/dx = (1/2) * x^(-1/2)
Step 2: Evaluate the derivative at x = c.
Substitute x = c into the derivative expression:
dy/dx = (1/2) * c^(-1/2)
This gives us the slope of the tangent at the point (c, sqrt(c)).
Step 3: Write the equation of the tangent line using the point-slope form, y - y1 = m(x - x1).
Using the point (c, sqrt(c)) and the slope [(1/2) * c^(-1/2)] we found in Step 2, we can substitute these values into the point-slope form:
y - sqrt(c) = (1/2) * c^(-1/2)(x - c)
Step 4: Simplify the equation.
To convert the equation to the slope-intercept form y = mx + b, we need to rearrange the equation to solve for y:
y = (1/2) * c^(-1/2)(x - c) + sqrt(c)
y = (1/2) * c^(-1/2)x - (1/2) * c^(-1/2)c + sqrt(c)
y = (1/2) * c^(-1/2)x - (1/2) * sqrt(c) + sqrt(c)
Simplifying further:
y = (1/2) * c^(-1/2)x + (1/2) * sqrt(c)
Therefore, the equation of the tangent line to the curve y = sqrt(x) at x = c is y = (1/2) * c^(-1/2)x + (1/2) * sqrt(c), where m is the slope given by (1/2) * c^(-1/2) and b is the y-intercept given by (1/2) * sqrt(c).