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# Class Note for MATH 1314 with Professor Gross at UH

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This 8 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Houston taught by a professor in Fall. Since its upload, it has received 13 views.

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Date Created: 02/06/15
M1314 lesson 3 2 Now suppose we move point Q closer to point P When we do this we ll get a better approximation of the slope of the tangent line Suppose we move point Q even closer to point P We get an even better approximation We are letting the distance betweenP and Q get smaller and smaller What does this sound like M1314 lesson 3 3 Now let s give these two points names We ll express them as ordered pairs Now we ll apply the slope formula to these two points This expression is called a difference quotient The last thing that we want to do is to let the distance betweenP and Q get arbitrarily small so we ll take a limit This gives us the definition of the slope of the tangent line Definition The slope of the tangent line to the graph of fat the point Px f x is given by hm fx hgt7 fx h gt0 1 provided the limit exists The difference quotient gives us the average rate of change We find the instantaneous rate of change when we take the limit of the difference quotient The derivative of f with respect to x is the function f 39 read fprime defined by f39x The domain of f39x is the set of all x for which the limit exists M1314 lesson 3 4 We can use the derivative of a function to solve many types of problems Once we have completed our study of the derivative we will be able to solve problems such as these Core Problems Core Problem 2 A ball is thrown straight up from the top of a building that is 64 feet high with an initial velocity of 96 feet per second so that its height in feet after t seconds is given by st 16t2 96t 64 1 Find the average velocity of the ball over the time intervals 2 3 2 25 2 21 2 Find the instantaneous velocity of the ball when t 2 3 Compare your results for questions 1 and 2 4 When does the ball hit the ground What is its velocity at the moment when its hits the ground Core Problem 3 The oxygen content of a pond 1 days after organic waste has been dumped into t2 41 4 the pond is given by f t 100 2 J t gt 0 percent of the normal level Sketch the t graph of the function and interpret your results Core Problem 4 A tennis racket manufacturer finds that the total daily cost of manufacturing x rackets per day is given by the function C x 400 4x 00001x2 The rackets can be sold at a price of p dollars where p 10 00004x Assume that all rackets that are manufactured can be sold Find the daily level of production that will yield a maximum profit The Four Step Process for Finding the Derivative Now that we know what the derivative is we need to be able to find the derivative of a function We ll use an algebraic process to do so We ll use a FourStep Process to find the derivative The steps are as follows 1 Find fxh 2 Find fxh fx fxhfx 3 Form the difference quotient h 4 Find the limit of the difference quotient as h gets close to 0 lim hgt0 fxhfx h v fxh fx Sofx lg11 h M1314 lesson 3 Example 1 Find the rule for the slope of the tangent line for the function f x 3x 2 Step 1 Step 2 Step 3 Step 4 Example 2 Use the fourstep process to nd the derivative of f x x2 Step 1 Step 2 Step 3 Step 4 M1314 lesson 3 Example 3 Find the derivative of f x x Example 4 Find the derivative if f x 4x2 2x 3 M1314 lesson 3 7 Example 5 Suppose f x x2 2x 5 Find the average rate of change of f over the interval 2 5 We can now answer Core Problem 2 Problem 2 A ball is thrown straight up from the top of a building that is 64 feet high with an initial velocity of 96 feet per second so that its height in feet after tseconds is given by st 16t2 96t 64 1 Find the average velocity of the ball over the time intervals 2 3 2 25 2 21 2 Find the instantaneous velocity of the ball when t 2 3 Compare your results for questions 1 and 2 4 When does the ball hit the ground What is its velocity at the moment when its hits the ground M1314 lesson 3 From this lesson you should be able to Explain what a derivative is State the limit de nition of the derivative Use the fourstep process to nd the derivative of a polynomial function Find the average rate of change M1314 lesson 3 1 Math 1314 Lesson 3 The Derivative The Limit De nition of the Derivative We now address the rst of the two questions of calculus the tangent line question We are interested in nding the slope of the tangent line at a speci c point J We could attempt to answer this question by graphing the function and its tangent line at the point of interest However with many functions we d get an approximation at best We need a way to nd the slope of the tangent line analytically for every problem that will be exact every time We can draw a secant line across the curve then take the coordinates of the two points on the curve P and Q and use the slope formula to approximate the slope of the tangent line Consider this function

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