Final answer:
To find the equation for the tangent to the curve at the point (4,8), use the derivative to find the slope of the tangent and then use the point-slope form of a line to write the equation. The equation of the tangent is y = 47x - 180. Sketch the curve y=47x and the tangent line together.
Step-by-step explanation:
To find the equation for the tangent to the curve at the point (4,8), we need to find the slope of the tangent and then use the point-slope form of a line to write the equation.
Step 1:
To find the slope of the tangent, we can use the derivative of the curve. The derivative of y=47x is dy/dx = 47.
Step 2:
Using the point-slope form of a line, we can write the equation of the tangent as y - y1 = m(x - x1), where (x1, y1) is the point of tangency and m is the slope of the tangent. Plugging in the given values, we get y - 8 = 47(x - 4).
Step 3:
Simplifying the equation, we have y - 8 = 47x - 188, y = 47x - 180.
To sketch the curve and the tangent together, plot the points (4,8) and then graph the curve y=47x. Finally, draw the straight line given by the equation y = 47x - 180 as the tangent.