Calculus 2.1 2.1 The Derivative A. Slope Review (Section 1.3) 1. Slope Formula – a. Dir: Find the slope of the line that passes through ( –5 , 2) and ( 1 , 12) 2. Slope-Intercept: a. Dir: Find the slope and intercepts. Then find the equation for the line. Slope: Intercepts: Equation: 1 Calculus 2.1 3. Point-Slope Formula: a. Dir: Write the equation for the line that passes through (2 , –4) with slope 1/3. B. Slope of a Secant (Average Rate of Change) 1. Use the graph below. Profit Yr 2, Profit = $0 million (millions Yr 5, Profit = $3.6 million of dollars) Time (years) a. Dir: Compute the slope of the secant line joining the points were x = 2 and x = 5 2 Calculus 2.1 2 2. Use fx()= x− 3 x a. Dir: Compute the slope of the secant line joining the points on the graph at x = –1 and x = 2. b. Dir: Compute the slope of the secant line joining the points on the graph at x = 1 and x = 3. 3 Calculus 2.1 C. Difference Quotient (Slope of Secant Line = Average Rate of Change) 1. Find the slope of the secant line joining the points on the graph at x and x+h. Coordinates of the first point: ( , ) Coordinates of the second point: ( , ) Slope: 4 Calculus 2.1 The formula for the difference quotient is: a. Note: fx()()()+ h≠ fx+ fh 2. Dir: find the difference quotient of f, fx()()+ h− fx namely h a. fx()=+ 3 x 2 5 Calculus 2.1 2 b. fx()= 5 x 2 c. fx()= x− 3 x 6 Calculus 2.1 D. Derivative (Slope of the Tangent Line = Instantaneous Rate of Change) 1. Concept Look at the point x = 2, and consider the secant line with h =1. The slope of secant line is -0.768. Now look at the point x = 2, and consider the secant line with h = 0.1. The slope of secant line is -0.461. 7 Calculus 2.1 Now look at the point x = 2, and consider the secant line with h = 0.01. The slope of secant line is -0.421. 2. The derivative of the function at x = 2 is the slope of the line you get by zooming in on the graph at x = 2. That line is called the tangent line. The slope of the tangent line is the limit of the slopes of the secant lines from the point on the graph at x = 2 to the point on the graph at x = 2 + h. Definition: 8 Calculus 2.1 3. Dir: Use calculus to compute the slope of the tangent line to the graph fx()for the given x-values. (Use the definition of derivative) a. fx()= 7 x2 at x = 1 9 Calculus 2.1 b. fx()= 4 x− 1 at x = 3 E. Derivative Terminology 1. If you are asked to find fx′(), that is the same as being asked to find the derivative. 2. fx′() can also be written as 3. So when you were asked to find f ′(3) that would be written as 10 Calculus 2.1 F. Using The Derivative 1. Dir: Compute the derivative of the given function and find the equation of the tangent line to its graph at the point for the given x-value. a. fx()= 7 x2 at x = 1 11 Calculus 2.1 G. Functions used in Economics (textbook page 5) 1. Total Cost: C(x) represents the total cost to produce x units of the commodity. Total Revenue: R(x) represents the total revenue gained by the sale of x units of the commodity. R(x) = quantity ! price = x ! p(x) Total Profit: P(x) represents the total profit obtained by the production and sales of x units of the commodity. P(x) = Revenue ! Cost = R(x) ! C(x) 2. The demand function p(x) and the total cost function C(x) for a particular commodity are given in terms of the level of production x. 2 p(x) = !0.02x + 29 C(x) = 1.43x + 18.3x + 15.6 a. Express the total revenue R as a function of x. R(x) = b. Express total profit P as a function of x. P(x) = 12 .