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# Class Note for MATH 1314 with Professor Heeth at UH

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This 6 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 17 views.

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Date Created: 02/06/15

Example 3 Find the equation of the line tangent to x 5 7 2x 7 3x2 when x 2 Example 4 Find the equation of the line tangent to x 1n2x 7 x2 when x 1 Horizontal Tangent Lines etc Some other basic applications involve nding where the slope of the tangent line is equal to a given number Example 5 Find all values of x for which the line tangent to x x3 7 4x2 4x 9 is horizontal Example 6 Find all values of x for which the slope of the line tangent to x 3x 7 21nx is equal to 0 Example 7 Find all values of x for which the slope of the line tangent to x x3 7 4x2 7 5x 7 is 3 Rates of Change Sometimes we re interested in nding the rate of change of a function at a specific point Example 8 A Chamber of Commerce commissions a study on population growth indicating that the city s population is growing according to the function Pt 40000 50f 150t where P represents the population tyears from now How fast is the population increasing in 4 years Example 9 Developers of a masterplanned community estimate that the population in thousands of the community I years from now will follow the function Pt 25t2 125t200 Find the rate at With the population 1s changing With respect to t1me What t2 5t 40 will be the population after 10 years At what rate will the population be changing when t 10 Velocity and Acceleration A common use of rate of change is to describe the motion of an object The function of the position of the object is with respect to time so it is usually a function of t instead of x Ifthe object changes position over time we can compute its rate of change which we refer to as velocity We can find either the average rate of change or the instantaneous rate of change depending on the question posed The average velocity on the interval x x h will be the difference quotient fx h fx h The instantaneous velocity at the point x c will be the derivative of the position function evaluated at c Velocity can be positive negative or zero If you throw a rock up in the air its velocity will be positive while it is moving upward and will be negative while it is moving downward We refer to the absolute value of velocity as speed When you accelerate while driving you are increasing your speed This means that you are changing your rate of change Acceleration then is the derivative of velocity 7 the rate of change of your rate of change It follows that the second derivative of a position functions gives an acceleration function So if position is given by ft and instantaneous velocity is given by vt f t then acceleration is given by at f 39t Example 10 The distance 3 in feet covered by a car I seconds after starting from rest is given by the function st t3 12t2 36L a Find the average velocity over the interval 1 2 b Find the instantaneous velocity when t 2 seconds c Find the acceleration when t 2 seconds From this lesson you should be able to Find the slope of a tangent line at a point Write an equation of a tangent line at a point Determine values of x for which the rst derivative is zero ie the tangent line is horizontal Determine values of x for which the rst derivative is some constant k Solve problems involving velocity and acceleration Solve problems involving other rates of change Math 1314 Lesson 8 Some Applications of the Derivative Equations of Tangent Lilies The rst applications of the derivative involve nding the slope of the tangent line and writing equations of tangent lines Example 1 Find the slope ofthe line tangent to fx x2 5x 15 at the point 2 3 Example 2 Find the equation of the line tangent to x 4x 7 2x2 at the point 3 6

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