Equation of Tangent Lines

Front

What is the definition of a tangent line?

Back

A tangent line to a curve at a given point is a straight line that touches the curve at that point and has the same slope as the curve at that point.

Front

How do you find the slope of a tangent line to a function at a specific point?

Back

The slope of the tangent line at a specific point can be found by calculating the derivative of the function at that point.

Front

What is the derivative of f(x) = -1(x + 12)^2 + 15?

Back

f'(x) = -2(x + 12).

Front

Calculate the derivative of f(x) = -1(x + 12)^2 + 15 at x = -9.

Back

f'(-9) = -2(-9 + 12) = -2(3) = -6.

Front

What is the point-slope form of a line?

Back

The point-slope form of a line is given by y - y1 = m(x - x1), where m is the slope and (x1, y1) is a point on the line.

Front

Using the point-slope form, write the equation of the tangent line to f(x) at x = -9.

Back

Using the point (-9, f(-9)) and slope -6, the equation is y - 15 = -6(x + 9).

Front

What is the value of f(-9) for the function f(x) = -1(x + 12)^2 + 15?

Back

f(-9) = -1(-9 + 12)^2 + 15 = -1(3)^2 + 15 = -9 + 15 = 6.

Tags

CCSS.HSF.IF.A.2

Front

What is the final equation of the tangent line to f(x) = -1(x + 12)^2 + 15 at x = -9?

Back

The final equation is y = -6x - 48.

Front

What does it mean if a tangent line does not exist at a point?

Back

A tangent line does not exist at a point if the function is not differentiable at that point, such as at a cusp or vertical tangent.

Front

What is the significance of the derivative in relation to tangent lines?

Back

The derivative represents the instantaneous rate of change of a function, which corresponds to the slope of the tangent line at a given point.