Derivatives and Equations of Tangent Lines!

Front

What is the derivative of a function?

Back

The derivative of a function measures how the function's output value changes as its input value changes. It represents the slope of the tangent line to the function's graph at a given point.

Front

How do you find the slope of the tangent line at a specific point on a curve?

Back

To find the slope of the tangent line at a specific point, calculate the derivative of the function and then evaluate it at the given x-coordinate.

Front

What is the equation of a tangent line?

Back

The equation of a tangent line at a point (a, f(a)) can be expressed as: y - f(a) = f'(a)(x - a), where f'(a) is the slope of the tangent line.

Front

What does it mean for a tangent line to be horizontal?

Back

A tangent line is horizontal when its slope is zero, which occurs at points where the derivative of the function is equal to zero.

Front

Given f(x) = 2x^3 - 4x + 1, find f'(x).

Back

f'(x) = 6x^2 - 4.

Front

Given r(x) = 2x^2 + 5x - 3, find the slope of the tangent line at x = 2.

Back

The slope of the tangent line at x = 2 is 9.

Front

What is the significance of the first derivative test?

Back

The first derivative test helps determine the local maxima and minima of a function by analyzing the sign of the derivative.

Front

How do you determine where a function has a horizontal tangent?

Back

Set the derivative of the function equal to zero and solve for x. The corresponding y-values can then be found by substituting x back into the original function.

Front

What is the derivative of f(x) = 3x^2 - 2x + 5?

Back

f'(x) = 6x - 2.

Front

What is the point-slope form of a linear equation?

Back

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