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# Class Note for MATH 1314 at UH

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COURSE
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KARMA
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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 15 views.

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Date Created: 02/06/15
Mach 1314 L essun 1o App licztinns uf Lhe First Derivative Demrmining Lhe Inmrvzls un Which a FuncLiun is Increasing ur Decreasing me the Graph urf De nitinn A function is increasing on an inta val a b if for any two numbers x and x in a b frgt lt fax whmeva r lt r A function is decreasing on an interval a b if for any two numbers r and r in ab frgt gtfx2 whmeva r ltX2 1n otha words ithey values are getting bigger as we move from left to right across the gaph othe function the function is increasing Ithey are getting smaller then the function is decreasing Example 1 You are given the gaph of x State the intervaKs on whichis increasing and the inta39vaKs on whichis decreasing 25 20 The rate of change of a function at a point is given by the daivative ofthe function aLLhaL point So we can use the derivative 10 deiermine whae a function is incieasing and whae a function is decreasing Let s look at the slopes of some lines that are tangent to this gmph at various points y Here are some generalizations At a point Where the derivative is positive a inction is increasing At a point Where the derivative is negative a inction is decreasing Determining the Intervals on Which a Function is Increasing or Decreasing By Finding the Derivative and Analyzing is Sign We can also determine Where a function is increasing and Where it is decreasing algebraically Here s how Find the derivative of the function Determine all values of x for Which f 39 x 0 or is unde ned Use the values found in step 2 to divide the number line into open intervals Choose a test value c in each open interval Substitute each test value 5 into the derivative to determine the sign of f 39c Apply the following theorem 9 5quot gtP Nt Theorem a If f39x gt 0 for each value ofx in an interval a b thenfis increasing 011a b b If f39x lt 0 for each value ofx in an interval a b thenfis decreasing 011a b c If f39x 0 for each value ofx in an interval a b thenfis constant 011a b Example 2 Determine the intervals where f is increasing and the interva1s where f is decreasing if fxx4 2x2 4 Example 3 Determine the intervals where f is increasing and the interva1s where f is decreasing if 1 3 3 2 x ix 7x 2x 5 f 3 2 Example 4 Determine the intervals where f is increasing and the interva1s where f is decreasing if fx x4 12x2 Example 5 Determine the intervals where f is increasing and the intervals where f is decreasing if fx xe Example 6 Determine the intervals where f is increasing and the interva1s where f is decreasing if fx 1nx 3 Finding Relative Extrema The rst derivative can also help up nd the x coordinates of any high points or low points on the graph of a function This will allow us to nd the x coordinates of the peaks and valleys in the graphs These high points andor low points are called relative extrema of a function An extremum is called a relative local maximum if it is higher than the points located nearby An extremum is called a relative local minimum if it is lower than the points located nearby Definition A function f has a relative maximum at x c if there exists an open interval a b containing 0 such that f x S f c for all x in a b A function f has a relative minimum at x c if there exists an open interval a b containing 0 such that f x 2 f c for all x in a b To nd the relative extrema we must rst nd the critical points of the function Definition A critical point of a function f is any point x in the domain of f such that f 39x 0 or f39x does not exist Once we nd the critical points we can use a line test to determine whether the critical point gives us a maximum a minimum or neither The First Derivative Test To nd the relative extrema of a function 1 Determine the critical points off 2 Determine the sign of f x to the left and to the right of each critical point a if f x changes sign from positive to negative as we move across a critical point x c from left to right then f c is a relative maximum b if f x changes sign from negative to positive as we move across a critical point x c from left to right then f c is a relative minimum c if f x does not change sign as we move across a critical point x c from left to right then f c is not a relative extremum Example 7 Find the relative extrema if fx x3 3x2 24x 32 Example 8 Find the relative extrema if f x x4 3x3 Example 9 Find the relative extrema if f x xzex Example 12 After birth an infant normally will lose weight for a few days and then start gaining A model for the average W in pounds of infants over the first two week following birth is Wt 033t2 3974t 73032 0 St 14 where tis measured in days Find the intervals on which weight is expected to increase and the intervals on which weight is expected to decrease based on this model From this section you should be able to Explain what we mean by an increasing decreasing function and relative extremum State intervals where a function is increasing decreasing from a graph of the function State intervals where a polynomial exponential or logarithmic function is increasing decreasing algebraically Find relative extrema of a polynomial exponential or logarithmic lnction algebraically Solve word problems involving intervals where a function is increasing decreasing

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