Question

Find an equation of the tangent to the curve at the given point. Then graph the curve and the tangent. (Do this on paper. Your instructor may ask you to turn in this work.) x 4sin(t) y2t (0, 0)

Answer 1

To find the equation of the tangent to the curve at the point (0,0), we need to find the slope of the curve at that point and use it to write the equation of the tangent line. The slope of the tangent line is equal to the derivative of y with respect to x. The equation of the tangent line is y=1/2x.

Step-by-step explanation:

To find the equation of the tangent to the curve at the point (0,0), we need to find the slope of the curve at that point. The slope of a curve at a point is equal to the slope of the tangent line at that point.The equation of the curve is given by x=4sin(t) and y=2t. At t=0, x=0 and y=0, so the point (0,0) lies on the curve.

To find the slope of the tangent line at t=0, we need to find the derivative of y with respect to x and evaluate it at t=0.The derivative of y with respect to x is dy/dx = dy/dt / dx/dt = (dy/dt)/(dx/dt) = (2)/(4cos(t)) = 1/(2cos(t)).Evaluating the derivative at t=0, we get dy/dx = 1/(2cos(0)) = 1/2.So, the slope of the tangent line at the point (0,0) is 1/2.Therefore, the equation of the tangent line to the curve at (0,0) is y=1/2x.