Introduction to the Derivative

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Presentation transcript:

Introduction to the Derivative Lecture 6 Introduction to the Derivative chapter3 Average Rate of Change The Derivative Derivatives of Powers, Sums and Constant Multiples Marginal Analysis Limits and Continuity

The Derivative Function Enough all ready!!! The Derivative Function If f is a function, its derivative function is the function whose value is the derivative of f at x. Clap if you don’t want to ever calculate another Difference Quotient… Ok…let’s find a shortcut… and ENJOY Calculus for a change!

Shortcut: The Power Rule If ( n is any constant) Then Ex. Ex.

Differential Notation: Differentiation means “the derivative with respect to x. The derivative of y with respect to x is written Ex. If y = x5 then

Differentiation Rules Ex. Ex.

Differentiation Rules Ex. Ex.

Functions Not Differentiable at a Point Cusp Vertical tangent

The Derivative of the Absolute Value of x. Note the derivative doesn’t exist when x = 0. Ex.

Position versus Velocity A position function s(t) specifies the position (height) as a function of time t. The velocity function is the derivative of s(t). It measures the rate of change of position with respect to t. The Velocity Function:

Cost Functions A cost function specifies the total cost C as a function of the number of items x. The marginal cost function is the derivative of C(x). It measures the rate of change of cost with respect to x. The Marginal Cost Function: The marginal cost function approximates the change in actual cost of producing an additional unit.

Cost Functions Given a cost function C, the average cost function is given by:

Cost Functions Ex. The monthly cost function C, for x items is given by: Find the marginal cost and the average cost functions.

Cost Functions The marginal cost function: The average cost function:

Marginal Functions The Marginal RevenueFunction measures the rate of change of the revenue function. Approximates the revenue from the sale of an additional unit. The Marginal Profit Function measures the rate of change of the profit function. Approximates the profit from the sale of an additional unit.

Marginal Functions Given a revenue function, R(q), The Marginal Revenue Function is: Given a profit function, P(q), The Marginal Profit Function is:

Marginal Revenue and Profit The monthly demand for T-shirts is given by Where p denotes the wholesale unit price in dollars and q denotes the quantity demanded. The monthly cost for these T-shirts is $8 per shirt. 1. Find the revenue and profit functions. 2. Find the marginal revenue and marginal profit functions.

Marginal Functions 1. Find the revenue and profit functions. Revenue = qp Profit = revenue – cost

Marginal Functions 2. Find the marginal revenue and marginal profit functions. Marginal revenue = Marginal profit =