Explanation Given: The pressure function is given as P ( t ) = t 3 − 18 t 2 + 81 t ( 0 ≤ t ≤ 15 ) . Here, t is time in years. Formula used: Write the formula for derivative of power rule. If f ( x ) = x n for any number n , then the derivative. f ′ ( x ) = n x n − 1 If f ( x ) = k for any real number k , then the derivative is defined by the formula. f ′ ( x ) = 0 Calculation: Let consider the pressure function as y = P ( t ) . The steps of calculating local extrema in TI83 calculator is given below: Step 1: Press ON button and then press Y= button. Step 2: Then, enter the function Y = x 3 − 18 x 2 + 81 x , x is the time. Step 3: Now, press window button and set the window as X min = 0 , X max = 16 , X s c l = 2 , Y min = − 20 , Y max = 120 and Y s c l = 10 . Step 4: Press GRAPH button, to get the graph. Step 5: Press 2nd + TRACE + 3 , and then enter the left bound as 0 and right bound as 15, to get the local minima. Step 6: It displays the answer as x = 9 and y = 0 . Step 7: Press 2nd + TRACE + 4 , and then enter the left bound as 0 and right bound as 9, to get the local maxima. Step 6: It displays the answer as x = 2.9999994 and y = 108 . It displays the graph as; For minima, and, for maxima, The approximate value of local minima is ( 9 , 0 ) and the approximate value of local maxima is ( 3 , 108 ) . Thus, the local minima of the function P ( t ) = t 3 − 18 t 2 + 81 t ( 0 ≤ t ≤ 15 ) lies at the point ( 9 , 0 ) and its local maxima lies at the point ( 3 , 108 ) . Consider the function P ( t ) = t 3 − 18 t 2 + 81 t . Now, differentiate the above function with respect to t, to get; P ′ ( t ) = d d t ( t 3 − 18 t 2 + 81 t ) = 3 t 2 − 36 t + 81 Differentiate the function P ′ ( t ) , to get; P ″ ( t ) = d d t ( 3 t 2 − 36 t + 81 ) = 6 t − 36 Now, the inflection point is the value of t where the function P ″ ( t ) vanishes

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Chapter 12.6, Problem 33E

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To calculate: The approximate location of all local extrema and points of inflection of the function with the help of graphing calculator of the pressure function is given as P(t)=t3−18t2+81t (0≤t≤15). Here, t is time in years.

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Chapter 12.6, Problem 32E

Chapter 12.6, Problem 34E

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The approximate location of all local extrema and points of inflection of the function with the help of graphing calculator of the pressure function is given as P ( t ) = t 3 − 18 t 2 + 81 t ( 0 ≤ t ≤ 15 ) . Here, t is time in years.

Solution Summary: The author calculates the approximate location of all local extrema and points of inflection of the function with the help of graphing calculator.

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To calculate: The approximate location of all local extrema and points of inflection of the function with the help of graphing calculator of the pressure function is given as P(t)=t3−18t2+81t (0≤t≤15). Here, t is time in years.

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