Lesson A tricky Calculus problem on derivative and anti-derivative
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<H2>A tricky Calculus problem on derivative and anti-derivative</H2> <H3>Problem 1</H3>f(x) is a polynomial function, f '(x) + integral of f(x)dx = x ^4 + 13 x ^2 + 2. Find f(x). <B>Solution</B> <pre> We want to find f(x) as a polynomial f(x) = {{{a[0]*x^n}}} + {{{a[1]*x^(n-1)}}} + . . . + {{{a[n]}}}. Taking derivative decreases the degree of a polynomial by one unit. Taking antiderivative increases the degree of a polynomial by one unit. Since the sum f ' (x) + int f (x) dx is a polynomial of degree 4, ---------------------- it means that the sough polynomial f(x) is of degree 3: f(x) = ax^3 + bx^2 + cx + d. Then f ' (x) = {{{3a*x^2}}} + 2bx + c, int f(x) dx = {{{(a/4)x^4}}} + {{{(b/3)x3}}} + {{{(c/2)x^2}}} + dx + E. So, in the sum f ' (x) + int f(x) dx ---------------------- (a) coefficient at {{{x^4}}} is {{{a/4}}}, which gives an equation {{{a/4}}} = 1; hence a = 4. (b) coefficient at {{{x^3}}} is 0, which gives an equation {{{b/3}}} = 0; hence b = 0. (c) coefficient at {{{x^2}}} is 13, which gives an equation {{{3a+ c/2}}} = 13, or {{{3*4 + c/2}}} = 13 ---> {{{c/2}}} = 13 - 12 = 1 ---> c = 2. (d) coefficient at {{{x}}} is 0, which gives an equation 2b + d = 0, which implies 2*0 + d = 0; hence, d = 0. +------------------------------------------------------------+ | At this point, the problem is just solved to the end. | | a = 4; b = 0; c = 2; d = 0. | +------------------------------------------------------------+ The sough polynomial is f(x) = 4x^3 + 2x. <U>ANSWER</U> <U>CHECK</U>. The derivative is f ' (x) = {{{12x^2 + 2}}}. The anti-derivative is F(x) = {{{(4/4)x^4 + (2/2)x^2}}} = {{{x^4 + x^2}}}. The sum f ' (x) + F(x) = {{{(12x^2+2)}}} + {{{(x^4 + x^2)}}} = {{{x^4 + 13x^2 + 2}}}. ! correct ! </pre> My other lessons on Calculus word problems at this site are - <A HREF=https://www.algebra.com/algebra/homework/word/misc/A-ladder-foot-slides-on-the-ground-txt.lesson>A ladder foot slides on the ground</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Finding-rate-of-change-of-some-processes.lesson>Finding rate of change of some processes</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Find-the-derivative-of-a-function-defined-by-complicated-expression.lesson>Find the derivative of a function defined by complicated expression</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Taking-derivative-of-a-function-which-is-defined-implicitly.lesson>Taking derivative of a function, which is defined implicitly</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Find-the-derivative-of-a-function-satisfying-given-functional-equation.lesson>Find the derivative for a function satisfying given functional equation</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/09-add-Find-the-Range-of-f%28x%29-=-%285cos%28x%29%29-div-%28x-%2B-1%29%29cos%28x%29.lesson>Find the range of f(x) = (5*cos(x))/(x + 1)), x >=0</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Couple-of-non-standard-Calculus-problems.lesson>Couple of non-standard Calculus problems</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Finding-the-minimum-of-a-function-defined-on-a-curve-in-the-coordinate-plane.lesson>Finding the minimum of a function defined on a curve in a coordinate plane</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Tricky-solution-to-find-the-maximum-of-a-function-defined-by-a-complicated-expression.lesson>Tricky solution to find the maximum of a function defined by a complicated expression</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Maximize-the-area-of-a-trapezoid.lesson>Maximize the area of a trapezoid </A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Maximmize-the-volume-of-an-open-box.lesson>Maximize the volume of an open box</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Minimize-surface-area-of-a-rectangular-prizm-with-the-given-volume.lesson>Minimize surface area of a rectangular box with the given volume</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Minimize-the-cost-of-an-aquarium-with-the-given-volume.lesson>Minimize the cost of an aquarium with the given volume</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Minimize-surface-area-of-a-conic-paper-cup-with-the-given-volume.lesson>Minimize surface area of a conical paper cup with the given volume</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Find-the-volume-of-a-solid-body-obtained-by-rotation-the-area-about-an-axis.lesson>Find the volume of a solid obtained by rotation of some plane shape about an axis</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Finding-the-volume-of-a-solid-body-mentally.lesson>Finding the volume of a solid body mentally</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/OVERVIEW-of-my-lessons-on-Calculus-word-problems.lesson>OVERVIEW of my lessons on Calculus word problems</A> Use this file/link <A HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A> to navigate over all topics and lessons of the online textbook ALGEBRA-II.
