Chapter 3: Applications of Differentiation
Section 3.10: Antiderivatives
If g″x=4 x2−9 x−5, find the antiderivative gx for which g−1=3 and g3=−1.
The most general antiderivative of g″ is g′, which will contain an arbitrary constant such as c1. Neither condition can be applied at this point because they both must be applied to g, not g′. Hence, g, the most general antiderivative of g′, must be found, and this will contain (in addition to c1) a second arbitrary constant such as c2. The two given conditions can then be applied to this latter antiderivative, resulting in two equations in the two unknowns c1 and c2.
= 4 x33−9 x22−5 x+c1
The condition g−1=3 yields the equation 13+32−52−c1+c2=3 or c2−c1=11/3.
The condition g3=−1 yields the equation 27−812−452+3 c1+c2=−1 or 3 c1+c2=35.
Simultaneously solving c2−c1=11/3 and 3 c1+c2=35 gives c1=47/6 and c2=23/2, so that
The Maple solution given below sets g″x and g′x as atomic identifiers. An atomic identifier is a collection of symbols that do not ordinarily form a valid Maple name for a variable. The atomic identifier version "locks" all the symbols together into a single name so that the notation can be used as a valid Maple variable. In Maple 17, atomic identifiers assume a default (red) color to indicate their status.
In addition, the assignment operator (≔) has been used to assign values to the atomic identifiers so defined. The two atomic identifiers used here are created by first typing the notation in math mode, then selecting the complete symbol and pressing three keys: Control+Shift+A.
The AntiderivativePlot command is used to obtain g′x from g″x. For polynomials, choosing the left endpoint of the interval of definition to be zero guarantees that the antiderivative will be the appropriate one. Note that the additive constant must be appended to the antiderivative.
After gx is obtained, it is rendered a function of x by the Assign Function option in the Context Panel. This allows the two equations that define the constants c1 and c2 to be readily written. The Context Panel is used to solve these equations, and the values so found are transferred back to gx with the Evaluation template in the Expression palette.
g″x≔4 x2−9 x−5:
→assign as function
gxx=a|f(x) = 13⁢x4−32⁢x3−52⁢x2+476⁢x+232
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