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Fig 34
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EXAMPLE 32
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Sketch the graph of f (x) = x 3
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SOLUTION First observe that f (x) = 3x 2 Thus f 0 everywhere The function is always increasing Second observe that f (x) = 6x Thus f (x) < 0 when x < 0 and f (x) > 0 when x > 0 So the graph is concave down on the negative real axis and concave up on the positive real axis Finally note that the graph passes through the origin We summarize our ndings in the graph shown in Fig 35 You Try It: Use calculus to aid you in sketching the graph of f (x) = x 3 + x EXAMPLE 33
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Sketch the graph of g(x) = x + sin x
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SOLUTION We see that g (x) = 1 + cos x Since 1 cos x 1, it follows that g (x) 0 Hence the graph of g is always increasing Now g (x) = sin x This function is positive sometimes and negative sometimes In fact sin x is positive when k < x < (k + 1) , k odd
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CHAPTER 3 Applications of the Derivative
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Fig 35
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and sin x is negative when k < x < (k + 1) , k even So the graph alternates being concave down and concave up Of course it also passes through the origin We amalgamate all our information in the graph shown in Fig 36
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Fig 36
EXAMPLE 34
Sketch the graph of h(x) = x/(x + 1 )
CHAPTER 3 Applications of the Derivative
SOLUTION First note that the function is unde ned at x = 1 We calculate that h (x) = 1/((x + 1)2 ) Thus the graph is everywhere increasing (except at x = 1) We also calculate that h (x) = 2/((x + 1)3 ) Hence h > 0 and the graph is concave up when x < 1 Likewise h < 0 and the graph is concave down when x > 1 Finally, as x tends to 1 from the left we notice that h tends to + and as x tends to 1 from the right we see that h tends to Putting all this information together, we obtain the graph shown in Fig 37
Fig 37
You Try It: Sketch the graph of the function k(x) = x EXAMPLE 35
Sketch the graph of k(x) = x 3 + 3x 2 9x + 6
x + 1
SOLUTION We notice that k (x) = 3x 2 + 6x 9 = 3(x 1)(x + 3) So the sign of k changes at x = 1 and x = 3 We conclude that k is positive when x < 3; k is negative when 3 < x < 1; k is positive when x > 3 Finally, k (x) = 6x + 6 Thus the graph is concave down when x < 1 and the graph is concave up when x > 1 Putting all this information together, and using the facts that k(x) when x and k(x) + when x + , we obtain the graph shown in Fig 38
CHAPTER 3 Applications of the Derivative
Fig 38
Maximum/Minimum Problems
One of the great classical applications of the calculus is to determine the maxima and minima of functions Look at Fig 39 It shows some (local) maxima and (local) minima of the function f
Fig 39
Notice that a maximum has the characterizing property that it looks like a hump: the function is increasing to the left of the hump and decreasing to the right of the hump The derivative at the hump is 0: the function neither increases nor decreases
CHAPTER 3 Applications of the Derivative
at a local maximum This is sometimes called Fermat s test Also, we see that the graph is concave down at a local maximum It is common to refer to the points where the derivative vanishes as critical points In some contexts, we will designate the endpoints of the domain of our function to be critical points as well Now look at a local minimum Notice that a minimum has the characterizing property that it looks like a valley: the function is decreasing to the left of the valley and increasing to the right of the valley The derivative at the valley is 0: the function neither increases nor decreases at a local minimum This is another manifestation of Fermat s test Also, we see that the graph is concave up at a local minimum Let us now apply these mathematical ideas to some concrete examples EXAMPLE 36