More Rules of DIFFERENTIATION Finding Derivatives This template can be used as a starter file to give updates for project milestones. Sections Sections can help to organize your slides or facilitate collaboration between multiple authors. On the Home tab, under Slides, click Section, and then click Add Section. Notes Use the Notes pane for delivery notes or to provide additional details for the audience. You can see these notes in Presenter View during your presentation. Keep in mind the font size (important for accessibility, visibility, videotaping, and online production) Coordinated colors Pay particular attention to the graphs, charts, and text boxes. Consider that attendees will print in black and white or grayscale. Run a test print to make sure your colors work when printed in pure black and white and grayscale. Graphics, tables, and graphs Keep it simple: If possible, use consistent, non-distracting styles and colors. Label all graphs and tables.

Differentiation Rules REVIEW Constant Function Rule Power Rule Constant Multiple Rule Sum & Difference Rule What is the project about? Define the goal of this project Is it similar to projects in the past or is it a new effort? Define the scope of this project Is it an independent project or is it related to other projects? * Note that this slide is not necessary for weekly status meetings

Derivatives of Products and Quotients We will use a new Rule to differentiate products and quotients Product Rule Quotient Rule What is the project about? Define the goal of this project Is it similar to projects in the past or is it a new effort? Define the scope of this project Is it an independent project or is it related to other projects? * Note that this slide is not necessary for weekly status meetings

Product Rule The rule states: The derivative of the first function times the second function PLUS the derivative of the second function times the first function. How it looks… * If any of these issues caused a schedule delay or need to be discussed further, include details in next slide.

Product Rule EXAMPLE: * If any of these issues caused a schedule delay or need to be discussed further, include details in next slide.

Product Rule EXAMPLE: * If any of these issues caused a schedule delay or need to be discussed further, include details in next slide.

Product Rule EXAMPLE: * If any of these issues caused a schedule delay or need to be discussed further, include details in next slide.

Product Rule EXAMPLE: * If any of these issues caused a schedule delay or need to be discussed further, include details in next slide.

Product Rule EXAMPLE: Find the equation of the line tangent to the graph of f(x) at x = 3 Find the value of x where the tangent is horizontal. * If any of these issues caused a schedule delay or need to be discussed further, include details in next slide.

Product Rule EXAMPLE: The tangent line is y = -2x – 18 Find the equation of the line tangent to the graph of f(x) at x = 3 The tangent line is y = -2x – 18 * If any of these issues caused a schedule delay or need to be discussed further, include details in next slide.

Product Rule EXAMPLE: Find the value of x where the tangent is horizontal. * If any of these issues caused a schedule delay or need to be discussed further, include details in next slide.

Product Rule EXAMPLE: One hour after a vampire has drained a victim, x milligrams of a particular virus is transmitted to the victim based on body temperature change. The change in body temperature T(x), in °F, is given by: Find T’(x) Find T’(1), T’(3) & T’(6). * If any of these issues caused a schedule delay or need to be discussed further, include details in next slide.

Product Rule EXAMPLE: * If any of these issues caused a schedule delay or need to be discussed further, include details in next slide.

Product Rule EXAMPLE: Find T’(1), T’(3) & T’(6). * If any of these issues caused a schedule delay or need to be discussed further, include details in next slide.

QUOTIENT Rule The rule states: The derivative of the TOP function times the Bottom function MINUS the derivative of the Bottom function times the TOP function all divided by the Bottom function squared. How it looks… * If any of these issues caused a schedule delay or need to be discussed further, include details in next slide.

QUOTIENT Rule EXAMPLE: * If any of these issues caused a schedule delay or need to be discussed further, include details in next slide.

QUOTIENT Rule EXAMPLE: * If any of these issues caused a schedule delay or need to be discussed further, include details in next slide.

QUOTIENT Rule EXAMPLE: * If any of these issues caused a schedule delay or need to be discussed further, include details in next slide.

QUOTIENT Rule EXAMPLE: * If any of these issues caused a schedule delay or need to be discussed further, include details in next slide.

QUOTIENT Rule EXAMPLE: Since Buffy came to Sunnydale, the total sales of wooden stakes t months after her arrival are given by: Find S’(t) Find S(10) and S’(10), then write an interpretation of the results. Estimate the total sales after 11 months. * If any of these issues caused a schedule delay or need to be discussed further, include details in next slide.

QUOTIENT Rule EXAMPLE: Since Buffy came to Sunnydale, the total sales of wooden stakes t months after her arrival are given by: Find S’(t) * If any of these issues caused a schedule delay or need to be discussed further, include details in next slide.

QUOTIENT Rule EXAMPLE: Since Buffy came to Sunnydale, the total sales (in $ dollars) of wooden stakes t months after her arrival are given by: Find S(10) and S’(10), then write an interpretation of the results. Sales after 10 months total $125. Sales increase monthly at a rate of $18.75. * If any of these issues caused a schedule delay or need to be discussed further, include details in next slide.

QUOTIENT Rule EXAMPLE: Estimate the total sales after 11 months. Since Buffy came to Sunnydale, the total sales of wooden stakes t months after her arrival are given by: Estimate the total sales after 11 months. Sales after 11 months are approximately $143.75. * If any of these issues caused a schedule delay or need to be discussed further, include details in next slide.

Quotient Rule EXAMPLE: According to economic theory, the demand x for a quantity in a free market decreases as the price p increases. Suppose that the number x of HDTVs people are willing to buy per week from a retail chain at a price of $p is given by: Find Find and interpret when p = $40. Estimate demand when the price is $41. * If any of these issues caused a schedule delay or need to be discussed further, include details in next slide.

Quotient Rule EXAMPLE: According to economic theory, the demand x for a quantity in a free market decreases as the price p increases. Suppose that the number x of HDTVs people are willing to buy per week from a retail chain at a price of $p is given by: * If any of these issues caused a schedule delay or need to be discussed further, include details in next slide.

Quotient Rule EXAMPLE: According to economic theory, the demand x for a quantity in a free market decreases as the price p increases. Suppose that the number x of HDTVs people are willing to buy per week from a retail chain at a price of $p is given by: At $40, the demand for HDTVs is 800 in weekly sales. However, at the same price level, demand is decreasing at a rate of 16 per week. * If any of these issues caused a schedule delay or need to be discussed further, include details in next slide.

Quotient Rule EXAMPLE: According to economic theory, the demand x for a quantity in a free market decreases as the price p increases. Suppose that the number x of HDTVs people are willing to buy per week from a retail chain at a price of $p is given by: At $41, the demand for HDTVs is 784 sales . * If any of these issues caused a schedule delay or need to be discussed further, include details in next slide.

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