Question

Consider the following curve.

y = 3x² − 7x + 1
Find the slope m of the tangent line at the point (3, 7).
m =
Find an equation of the tangent line to the curve at the point (3, 7).
y =

Answer 1

To find the slope of the tangent line at the point (3, 7), take the derivative of the given equation and plug in x = 3. The slope is 11. The equation of the tangent line is y = 11x - 26.

Step-by-step explanation:

To find the slope of the tangent line at the point (3, 7), we need to find the derivative of the given equation. Taking the derivative of y = 3x² - 7x + 1 gives us y' = 6x - 7. Plugging in x = 3 into the derivative equation gives us the slope of the tangent line, m = 6(3) - 7 = 11.

Therefore, the slope of the tangent line at the point (3, 7) is m = 11.

To find an equation of the tangent line at the point (3, 7), we can use the slope-intercept form of a line, y = mx + b, where m is the slope and b is the y-intercept.

Plugging in the values m = 11 and (x, y) = (3, 7) into the equation, we get y = 11x + b. To find the value of b, we can plug in the coordinates of the point (3, 7), which gives us 7 = 11(3) + b. Solving for b, we find that b = -26.

Therefore, an equation of the tangent line to the curve at the point (3, 7) is y = 11x - 26.