Question

Find an equation of the tangent line and normal line to the curve y=x*sqrt(x) at the point (1,1)

Answer 1

The equation of the tangent line to the curve y=x*sqrt(x) at the point (1,1) is y - 1 = (3/2)(x - 1), and the equation of the normal line at the same point is y - 1 = (-2/3)(x - 1).

Step-by-step explanation:

Finding the Tangent and Normal Lines to the Curve

To find an equation of the tangent line and normal line to the curve y=x*sqrt(x) at the point (1,1), we follow these steps:

1. First, find the derivative of the function, y', which will give us the slope of the tangent line at any point x.
2. Next, evaluate the derivative at x=1 to find the slope at the point (1,1).
3. With the slope of the tangent line, we can use the point-slope form to write the equation of the tangent line.
4. To find the equation of the normal line, we use the negative reciprocal of the tangent's slope and the same point (1,1).

The derivative of the function y=x*sqrt(x) is 3/2*x^(1/2). Evaluating the derivative at x=1 we get a slope of 3/2 for the tangent line. Now, using the point-slope form:

y - 1 = (3/2)(x - 1)

This is the equation of the tangent line. For the normal line, the slope is the negative reciprocal of 3/2, which is -2/3. Hence, the equation of the normal line is:

y - 1 = (-2/3)(x - 1)