CK12 Calculus - Single Variable
8. Ify= cscx, findy" (π/6). Use the definitionof the derivativeto provethat Answers 1.y= x cos x + sin x 3.y= 2 cos x sin x ...
And a final way of expressingthe chainrule is the easiestform to remember:Ifyis a functionofuanduis a functionofx, then Example1 ...
Example2: Whatis the slopeof the tangentline to the function that passesthroughpointx= 3? Solution: We can write This exampleill ...
= -5 sinu. (6x) = -30xsin(3x^2 - 1) Example5: Find for Solution: This exampleappliesthe chainrule twicebecausethereare severalfu ...
3. 4.f(x) = sin^3 x 5.f(x) = sinx 6.f(x) = sin^3 x^3 7.f(x) = tan(4x^5 ) 10.f(x) = (5x+ 8)^3 (x^3 + 7x)^13 Answers ...
ImplicitDifferentiation LearningObjectives A studentwill be able to: Find the derivativeof varietyof functionsby usingthe techn ...
whichagreeswith the previouscalculations.This secondmethodis calledtheimplicitdifferentiation method.You may wonderand say that ...
First we needto use implicitdifferentiationto finddy/dxand then substitutethe point(1, 2) into the derivative to find slope.Then ...
Use implicitdifferentiationto findd^2 y/dx^2 if Also find Whatdoesthe secondderivativerepresent? Solution: Solvingfordy/dx, Diff ...
in kinematics(the studyof motion),the speedof an object(y) signifiesthe changeof positionwith respect to time but acceleration(y ...
6. 7. 8. 9. Linearizationand Newton’s Method LearningObjectives A studentwill be able to: Approximatea functionby the methodof ...
So for valuesofxclosetox 0 , the valuesofyof this tangentline will closelyapproximatef(x). This givesthe approximation The Tange ...
Solution: Takingthe derivativeoff(x), we have , and This tells us that near the pointx= 1, the function approximatesthe liney= ( ...
Newton’s Method Whenfacedwith a mathematicalproblemthat cannotbe solvedwith simplealgebraicmeans,suchas findingthe rootsof the p ...
Noticethat this equationis mucheasierto solvethan Settingf(x) = 0 and solvingforx, we obtain, If you use a calculator, you will ...
Example3: Use Newton’s methodto find the rootsof the polynomial Solution: Usingthe recursionrelation, To help us find the first ...
Usingthe recursionrelationagainto findx 3 , we get We concludethat the solutionto the equation is about0.6836403. ReviewQuestion ...
c. d. e. Withoutusinga calculator, approximate(1.003)^99. Use Newton’s Methodto find the rootsofx^3 + 3 = 0. Use Newton’s Metho ...
3. Applicationsof Derivatives................................................................................................... ...
Simplifying,we have Equation1. So we haverelationshipsbetweenthe derivatives,and sincethe derivativesare rates,this is an exampl ...
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