The Second Fundamental Theorem of Calculus

In this topic:

(1) Proposition (Second Fundamental Theorem of Calculus) Let second fundamental theorem of calculus _gr_1.gif] be a continuous function on the interval second fundamental theorem of calculus _gr_2.gif] and let second fundamental theorem of calculus _gr_3.gif] be the function defined by the rule

second fundamental theorem of calculus _gr_4.gif]

Then second fundamental theorem of calculus _gr_5.gif] is the anitderivative of second fundamental theorem of calculus _gr_6.gif] on second fundamental theorem of calculus _gr_7.gif] that is,

second fundamental theorem of calculus _gr_8.gif]

on second fundamental theorem of calculus _gr_9.gif]

(2) Example (Second Fundamental Theorem of Calculus) Differentiate the function second fundamental theorem of calculus _gr_10.gif]

(3) Example (Second Fundamental Theorem of Calculus) Differentiate the function second fundamental theorem of calculus _gr_11.gif]

(4) Example (Second Fundamental Theorem of Calculus) Find second fundamental theorem of calculus _gr_12.gif] using the second fundamental theorem of calculus.

(5) Example (Second Fundamental Theorem of Calculus) Find second fundamental theorem of calculus _gr_13.gif] using the second fundamental theorem of calculus.

(6) Example (Second Fundamental Theorem of Calculus) Find an equation for the tangent line to the curve second fundamental theorem of calculus _gr_14.gif] at the point second fundamental theorem of calculus _gr_15.gif] where second fundamental theorem of calculus _gr_16.gif] and second fundamental theorem of calculus _gr_17.gif]

(7) Example (Second Fundamental Theorem of Calculus) Find an equation for the tangent line to the curve second fundamental theorem of calculus _gr_18.gif] at the point second fundamental theorem of calculus _gr_19.gif] where second fundamental theorem of calculus _gr_20.gif] and second fundamental theorem of calculus _gr_21.gif]

(8) Example (Second Fundamental Theorem of Calculus) Suppose second fundamental theorem of calculus _gr_22.gif] Find second fundamental theorem of calculus _gr_23.gif]  

(9) Example Let second fundamental theorem of calculus _gr_24.gif],  where second fundamental theorem of calculus _gr_25.gif] is a function whose graph is shown. Estimate second fundamental theorem of calculus _gr_26.gif] second fundamental theorem of calculus _gr_27.gif] second fundamental theorem of calculus _gr_28.gif] second fundamental theorem of calculus _gr_29.gif] and second fundamental theorem of calculus _gr_30.gif] Find the largest open interval on which second fundamental theorem of calculus _gr_31.gif] is increasing. Find the largest open interval on which second fundamental theorem of calculus _gr_32.gif] is decreasing. Identify any extrema of second fundamental theorem of calculus _gr_33.gif] Sketch a rough graph of second fundamental theorem of calculus _gr_34.gif]

    The second fundamental theorem allows us to take the derivative of a function defined in terms of a definite integral with a variable as an upper limit. It can also be used when the variable is the lower limit os integration by reversed the limits of integration and multiplying by negative one.

Proposition (Second Fundamental Theorem of Calculus) Let second fundamental theorem of calculus _gr_35.gif] be a continuous function on the interval second fundamental theorem of calculus _gr_36.gif] and let second fundamental theorem of calculus _gr_37.gif] be the function defined by the rule

second fundamental theorem of calculus _gr_38.gif]

Then second fundamental theorem of calculus _gr_39.gif] is the anitderivative of second fundamental theorem of calculus _gr_40.gif] on second fundamental theorem of calculus _gr_41.gif] that is,

second fundamental theorem of calculus _gr_42.gif]

on second fundamental theorem of calculus _gr_43.gif]

Example (Second Fundamental Theorem of Calculus) Differentiate the function

second fundamental theorem of calculus _gr_44.gif]

    Solution.  Since second fundamental theorem of calculus _gr_45.gif] is continuous on the interval second fundamental theorem of calculus _gr_46.gif] where second fundamental theorem of calculus _gr_47.gif] is any real number greater than 6, we see that the second fundamental theorem applies to the interval second fundamental theorem of calculus _gr_48.gif] and so,

second fundamental theorem of calculus _gr_49.gif]

second fundamental theorem of calculus _gr_50.gif]

Example (Second Fundamental Theorem of Calculus) Differentiate the function

second fundamental theorem of calculus _gr_51.gif]

    Solution.  Since

second fundamental theorem of calculus _gr_52.gif]
    
and second fundamental theorem of calculus _gr_53.gif] is continuous on the interval second fundamental theorem of calculus _gr_54.gif] where second fundamental theorem of calculus _gr_55.gif] is any real number greater than 12, we see that the second fundamental theorem applies to the interval second fundamental theorem of calculus _gr_56.gif] and so,

second fundamental theorem of calculus _gr_57.gif]

second fundamental theorem of calculus _gr_58.gif]

Example (Second Fundamental Theorem of Calculus) Find second fundamental theorem of calculus _gr_59.gif] using the second fundamental theorem of calculus.

    Solution. Let second fundamental theorem of calculus _gr_60.gif] and we will use the second fundamental theorem of calculus,

second fundamental theorem of calculus _gr_61.gif]

second fundamental theorem of calculus _gr_62.gif]

second fundamental theorem of calculus _gr_63.gif]

second fundamental theorem of calculus _gr_64.gif]

We can check  

second fundamental theorem of calculus _gr_65.gif]

Thus,  

second fundamental theorem of calculus _gr_66.gif]
second fundamental theorem of calculus _gr_67.gif]

Example (Second Fundamental Theorem of Calculus) Find second fundamental theorem of calculus _gr_68.gif] using the second fundamental theorem of calculus.

    Solution.   

Example (Second Fundamental Theorem of Calculus) Find an equation for the tangent line to the curve second fundamental theorem of calculus _gr_69.gif] at the point second fundamental theorem of calculus _gr_70.gif] where second fundamental theorem of calculus _gr_71.gif] and second fundamental theorem of calculus _gr_72.gif]

    Solution.   

Example (Second Fundamental Theorem of Calculus) Find an equation for the tangent line to the curve second fundamental theorem of calculus _gr_73.gif] at the point second fundamental theorem of calculus _gr_74.gif] where second fundamental theorem of calculus _gr_75.gif] and second fundamental theorem of calculus _gr_76.gif]

    Solution.   

Example (Second Fundamental Theorem of Calculus) Suppose second fundamental theorem of calculus _gr_77.gif] Find second fundamental theorem of calculus _gr_78.gif]

    Solution.   

Example (Second Fundamental Theorem of Calculus) Let second fundamental theorem of calculus _gr_79.gif],  where second fundamental theorem of calculus _gr_80.gif] is a function whose graph is shown. Estimate second fundamental theorem of calculus _gr_81.gif] second fundamental theorem of calculus _gr_82.gif] second fundamental theorem of calculus _gr_83.gif] second fundamental theorem of calculus _gr_84.gif] and second fundamental theorem of calculus _gr_85.gif] Find the largest open interval on which second fundamental theorem of calculus _gr_86.gif] is increasing. Find the largest open interval on which second fundamental theorem of calculus _gr_87.gif] is decreasing. Identify any extrema of second fundamental theorem of calculus _gr_88.gif]

second fundamental theorem of calculus _gr_89.gif]

    Solution.  First,
    
second fundamental theorem of calculus _gr_90.gif]

second fundamental theorem of calculus _gr_91.gif]

second fundamental theorem of calculus _gr_92.gif]

second fundamental theorem of calculus _gr_93.gif]

second fundamental theorem of calculus _gr_94.gif]

The largest open iterval where second fundamental theorem of calculus _gr_95.gif] is increasing is second fundamental theorem of calculus _gr_96.gif] and the largest open interval on which second fundamental theorem of calculus _gr_97.gif] is decreasing is second fundamental theorem of calculus _gr_98.gif]Since second fundamental theorem of calculus _gr_99.gif] is continuous on second fundamental theorem of calculus _gr_100.gif] the second fundamental theorem yields second fundamental theorem of calculus _gr_101.gif] defined on second fundamental theorem of calculus _gr_102.gif] Therefore, a maximium occurs when second fundamental theorem of calculus _gr_103.gif] which is at second fundamental theorem of calculus _gr_104.gif] second fundamental theorem of calculus _gr_105.gif]

Cite this as:
Second Fundamental Theorem Of Calculus
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
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