Chapter 3 Introduction to the Derivative Sections 3. 5, 3. 6, 4 Chapter 3 Introduction to the Derivative Sections 3.5, 3.6, 4.1 and 4.2

Introduction to the Derivative Average Rate of Change The Derivative

Average Rate of Change The change of f (x) over the interval [a,b] is The average rate of change of f (x) over the interval [a , b] is Difference Quotient

Average Rate of Change Is equal to the slope of the secant line through the points (a , f (a)) and (b , f (b)) on the graph of f (x) secant line has slope mS

Average Rate of Change as h0 The average rate of change of f over the interval [a , b] can be written in two different ways:

Average Rate of Change as h0 We now look at the behavior of the average rate of change of f (x) as b a  h gets smaller and smaller, that is, we will let h tend to 0 (h0) and look for a geometric interpretation of the result. For this, we consider an example of values of and their corresponding secant lines.

Average Rate of Change as h0 Secant line tends to become the Tangent line

Average Rate of Change as h0 Secant line tends to become the Tangent line

Average Rate of Change as h0 Secant line tends to become the Tangent line

Average Rate of Change as h0 Secant line tends to become the Tangent line

Average Rate of Change as h0 Secant line tends to become the Tangent line

Average Rate of Change as h0 Secant line tends to become the Tangent line

Average Rate of Change as h0 Secant line tends to become the Tangent line Zoom in

Average Rate of Change as h0 Secant line tends to become the Tangent line

Average Rate of Change as h0 Secant line tends to become the Tangent line

Average Rate of Change as h0 We observe that as h approaches zero, The secant line through P and Q approaches the tangent line to the point P on the graph of f. and consequently, the slope mS of the secant line approaches the slope mT of the tangent line to the point P on the graph of f.

Instantaneous Rate of Change of f at x = a That is, the limiting value, as h gets increasingly smaller, of the difference quotient is the slope mT of the tangent line to the graph of the function f (x) at x = a. (slope of secant line)

Instantaneous Rate of Change of f at x = a This is usually written as tends to, or approaches That is, mS approaches mT as h tends to 0.

Instantaneous Rate of Change of f at x = a More briefly using symbols by writing and read as “the limit as h approaches 0 of ”

Instantaneous Rate of Change of f at x = a that is, the limit symbol indicates that the value of can be made arbitrarily close to mT by taking h to be a sufficiently small number.

Instantaneous Rate of Change of f at x = a Definition: The instantaneous rate of change of f (x) at x = a is defined to be the slope of the tangent line to the graph of the function f (x) at x = a. That is, Remark: The slope mT gives a precise indication of how fast the graph of f (x) is increasing or decreasing at x = a.

A Concrete Example Consider the function f (x)  – x2/4 + 9/4, whose graph is given below, at the points a  – 3 and a  – 1 At which of these two points is the function increasing faster? Intuition says at x  – 3 because we notice the graph is steeper at the point x  – 3. Why?

A Concrete Example Consider the function f (x)  – x2/4 + 9/4, whose graph is given below, at the points a  – 3 and a  – 1 To make this more obvious we zoom in Because our brain makes no distinction between the graph of f and the tangent line to the graph at the point in question.

A Concrete Example Consider the function f (x)  – x2/4 + 9/4, whose graph is given below, at the points a  – 3 and a  – 1 We see how the tangent line basically coincides with the graph of f near the point of contact. What is the slope of each line?

A Concrete Example Consider the function f (x)  – x2/4 + 9/4, whose graph is given below, at the points a  – 3 and a  – 1 That is, at the points of contact, the function is increasing as fast as its corresponding tangent line.

A Concrete Example Consider the function f (x)  – x2/4 + 9/4, whose graph is given below, at the points a  – 3 and a  – 1 We now verify the claims by explicitly computing for this function, the limit,

A Concrete Example For the function f (x)  – x2/4 + 9/4 at any point a we have

A Concrete Example Thus, for the function f (x)  – x2/4 + 9/4 at any point a we have Therefore, as h tends to 0, mS approaches – a/2  mT .

A Concrete Example For the function f (x)  – x2/4 + 9/4 at any point a we have Thus,

A Concrete Example Let us try some other values of a for the same function.

Remark The example clearly indicates that the slope mT of the tangent line is itself a function of the point a we choose. We need a better notation reflecting the fact that this object is a function of a

The Derivative The instantaneous rate of change mT at a is also called the derivative of f at x = a and it is denoted by f '(a). That is, f '(a) is read “f prime of a” mT = f '(a) = “slope of the tangent line to f (x) at a” = “instantaneous rate of change of f at a”

The Derivative The instantaneous rate of change mT at a is also called the derivative of f at x = a and it is denoted by f '(a). That is, In our previous example, the derivative of the function f (x)  – (x2 – 9)/4 at any point a is given by

Rates of Change Average rate of change of f over the interval [a, a+h] is Slope of the secant line through the points (a , f (a)) and (a , f (a+h)) Instantaneous rate of change of f at x=a is Slope of the tangent line at the point (a , f (a))

Tangent Line and Secant Line tangent line at a

Using the point-slope form of the line Equation of Tangent Line at x = a Using the point-slope form of the line tangent line at a

Terminology Finding the derivative of the function f is called differentiating f. If f '(a) exists, then we say that f is differentiable at x = a. For some functions f , f '(a) may not exist. In this case we say that the function f is not differentiable at x = a.

Quick Approximation of the Derivative Recall that f '(a) is the limiting value of the expression as we make h increasingly smaller. Therefore, we can approximate the numerical value of the derivative using small values of h. h = 0.001 often works. In this case,

Quick Approximation - Example Demand: The demand for an old brand of TV is given by where p is the price per TV set, in dollars, and q is the number of TV sets that can be sold at price p . Find q (190) and estimate q '(190) . Interpret your answers.

Solution

Geometric Interpretation At a  190, q(p) decreases as fast as the tangent line does, that is, at the rate mT  q '(190)  2.5 TVs/$ What does this means? It means that at the price of p  $190, the demand will decrease by 2.5 TV sets per dollar we increase the price. a  190 If we now set the price at $200, how many TV sets do we expect to sell?

Geometric Interpretation At a  190, the equation of the tangent line is y  2.5( p-190 )+500 Thus, when we set the price at p  $200, the line shows that we expect to sell y  475 TVs The actual number we expect to sell according to the demand function is q (200)  476.2  476 TVs which is very close to the prediction given by the tangent line. p  200

Now Recall The Example The slope mT  f '(a) of the tangent line depends on the point a we choose on the graph of f .

Chapter 4 Techniques of Differentiation with Applications Sections 4 Chapter 4 Techniques of Differentiation with Applications Sections 4.1 and 4.2

Techniques of Differentiation Derivatives of Powers, Sums and Constant asMultiples Marginal Analysis

The Derivative as a Function If f is a function, its derivative function is the function whose value at x is the derivative of f at x. That is, Notice that all we have done is substituted x for a in the definition of f '(a). This way, when we are done with the algebra, the answer will be given in terms of x.

The Derivative as a Function Example: Given the function

Geometric Verification At any point on the graph of f , the tangent line agrees with the graph of which already is a straight line of slope 1

The Derivative as a Function Example: Given the function

Geometric Verification x y=x2 -3 -2 -1 1 2 3 x y'=2x -3 -2 -1 1 2 3

The Derivative as a Function Example: Given the function

The Power Rule Example: Example: Example:

Verification when f (x)  x1/2 Example: Given the function

Leibniz’ d Notation For Derivatives Average rate of change Instantaneous rate of change means the same as

Differential Notation: Differentiation means “the derivative with respect to x”. The derivative of y  f (x) with respect to x is written or as Example:

Differential Notation: Differentiation Example: To find f '(2), the derivative of y  f (x) at x  2, we first need to find the function f '(x). In Leibniz’ notation f '(2) is denoted by the symbol

Differentiation Rules Example: Example:

Differentiation Rules Example:

Differentiation Rules Example:

More Examples Example: Find the derivative of First write f (x) as constant times a power

Example: Find the derivative of First write f (x) as

Functions Not Differentiable at a Point a) Vertical tangent x = a b) Cusp x = a c) f (a) = Undefined

Example: The Absolute Value of x.

Example: The cube root of x.

Introduction to the Derivative Average Rate of Change The Derivative Derivatives of Powers, Sums and Constant asMultiples Marginal Analysis

C(x)= “variable costs” + “fixed costs” Cost Functions A cost function specifies the total cost C as a function of the number of items x produced. Thus, C(x) is the cost of x items. The cost functions is made up of two parts: C(x)= “variable costs” + “fixed costs” The marginal cost function is the derivative C′(x) of C(x). It measures the instantaneous rate of change of cost with respect to x.

By definition the Marginal Cost Function is C′(x). The units of marginal cost are units of cost (say, $) per item. Interpretation: The marginal cost function C′(x) approximates the change in actual cost of producing an additional unit when x items are already being produced.

Interpretation of Marginal Cost The Marginal Cost Function is C′(x). Thus, for small h we have the quick approximation given by If we let h=1 (one more item being produced ), then

Interpretation of Marginal Cost That is, for h=1, we have Notice that C(x+1) - C(x) is the actual (exact) cost of producing one additional item when the production level is x. In practice, h=1 is considered to be small in relation to production level x, which in general is a large number.

Average Cost Functions Given a cost function C(x) , the average cost function given by:

Example The total monthly cost function C, in dollars, of producing x computer games is given by: Find the marginal cost function and use it to estimate the cost of producing the 1,001st game. Find the actual cost of producing the 1,001st game.

Solution The marginal cost function is given by: An estimate for the cost of producing the 1,001st game is That is, it costs an additional $18 to produce one more game when the production level is at 1,000 games a month

Solution The the actual cost of producing the 1,001st game

Example Continued The total monthly cost function C, in dollars, of producing x computer games is given by: Find the average cost per game if 1,000 games are manufactured. Find the average cost function.

Solution The average cost function:

More Marginal Functions The Marginal Revenue Function measures the rate of change of the revenue function. It approximates the revenue from the sale of an additional unit. The Marginal Profit Function measures the rate of change of the profit function. It approximates the profit from the sale of an additional unit.

More 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 Functions - Examples 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. Find the revenue and profit functions. Find the marginal revenue and marginal profit functions.

Solution Revenue = qp Profit = revenue – cost Find the revenue and profit functions. Revenue = qp Profit = revenue – cost

Solution Marginal profit Find the marginal revenue and marginal profit functions. Marginal revenue Marginal profit