Question

Consider the following curve.

y=3x213x+1
Find the slope m of the tangent line at the point (5, 11).
m =
24
Find an equation of the tangent line to the curve at the point (5, 11).
y =

Answer 1

The equation of the tangent line to the curve at the point (5, 11) with a slope of 24 is y = 24x - 109, derived using the point-slope form of the line equation.

Step-by-step explanation:

To find the equation of the tangent line to the curve at the point (5, 11), we first need to verify the given slope m = 24. This slope value is what we would obtain if we differentiated the function y with respect to x and evaluated the derivative at x = 5. Assuming the slope is correct, the next step is to use the point-slope form of the equation of a line, which is y - y1 = m(x - x1), where (x1, y1) is a point on the line (in this case, (5, 11)) and m is the slope of the tangent.

Plugging in our values:

y - 11 = 24(x - 5)

Simplifying, we get:

y = 24x - 120 + 11

y = 24x - 109

Therefore, the equation of the tangent line to the curve at the point (5, 11) is y = 24x - 109.