Calculus Derivatives and Tangent Lines

Derivatives and Tangent Lines

• The derivative of a function at a point gives the slope of the tangent line to the graph of the function at that point.
• The slope of the tangent line to the graph of f(x) = -3x(x+2)^2 at x = -1 is -3.

Limits and Derivatives

• The derivative of a function f(x) is defined as the limit of the difference quotient as h approaches 0:
• lim h->0 [f(x+h) - f(x)] / h
• The limit as h approaches 0 of (x+h - x) / h is x.

Derivatives and Function Properties

• If g'(1) = -3, then the function g(x) could be represented by equations that have a derivative equal to -3 when x = 1.
• The derivative of a polynomial function f(x) is negative when f(x) is decreasing.
• The derivative of a polynomial function f(x) changes from negative to positive when f(x) has a relative minimum.
• The derivative of a polynomial function f(x) is equal to zero at the points where f(x) has a horizontal tangent line.

Finding Derivatives

• The derivative of f(x) = (x^2 + 2x) / x is (3x + 1) / 2x.

The Derivative and the Shape of a Function

• If the derivative of a function f(x) is positive, then the function is increasing.
• If the derivative of a function f(x) is negative, then the function is decreasing.
• If the derivative of a function f(x) is zero, then the function has a horizontal tangent line, which may be a relative maximum, minimum, or neither.
• The slope of the normal line at a point is the negative reciprocal of the slope of the tangent line at that point.

Finding Relative Extrema (Maxima and Minima)

• If the derivative of a function f(x) changes from positive to negative at a point x = a, then f(x) has a relative maximum at x = a.
• If the derivative of a function f(x) changes from negative to positive at a point x = a, then f(x) has a relative minimum at x = a.
• The graph of f(x) = 2x - 3cosx has a relative maximum at x = 1.895 and a relative minimum at x = 4.937.

Determining Intervals of Increase and Decrease

• The function f(x) is increasing on the intervals (0, 1.895) and (4.937, 2π).
• The function f(x) is decreasing on the interval (1.895, 4.937).