CK12 Calculus - Single Variable
Noticethat the points(t 0 ,x 0 ) and (t 1 ,x 1 ) lie on the position-versus-timecurve,as Figure1 shows.This expressionis also th ...
InstantaneousRateof ChangeThe instantaneousrate of changeofx=f(t) at the timet 0 is the slopemtanof the tangentline at the timet ...
Fromthe exampleabove,we foundthatf(x) = 2x, so This meansthat the instantaneousrate of changeis negative.That is,yis decreasin ...
b. Whatis the instantaneousvelocityof the particleatt= 2 nanoseconds? Answers a. , b. , c. , d. a. , b. , c. , d. 2x, 12. a. , ...
where Substitutinginto the derivativeformula, Example2: Find the derivativeof and the equationof the tangentline atx 0 = 1. Solu ...
Forx 0 = 1, we can findy 0 by simplysubstitutingintof(x). Thusthe equationof the tangentline is Notation Calculus,just like all ...
Doesnot exist then the derivativef(x) also fails to existas x → x 0. The followingexamplesshowfour caseswhere the derivativefail ...
Manyfunctionsin mathematicsdo not havecorners,cusps,verticaltangents,or jumpdiscontinuities.We call themdifferentiablefunctions. ...
Fromwhatwe havelearnedalreadyaboutdifferentiability, it will not be difficult to showthat continuityis an importantconditionfor ...
and Findf(0) andf(x). Answers 1. 2. 3. 4. 5. 6. 10 Hint: Take the limit from both sides. Hint: Take the limit from both sides. ...
Proof: Example1: Iff(x) = 16 for allx, thenf(x) = 0 for all x. We can also writed/dx (16) = 0. The PowerRuleIfnis a positiveinte ...
Example4: Derivativesof Sumsand Differences Iffandgare two differentiablefunctionsatx, then and In simplernotation, Example5: Ex ...
Keepin mindthat Example7: Find for Solution: Thereare two methodsto solvethis problem.One is to multiplythe productand then use ...
Example8: Finddy/dxfor Solution: Example9: At whichpoint(s)doesthe graphof havea horizontaltangentline? Solution: Sincethe slope ...
fourth,fifth and higherderivativesof f. Theyare denotedby f ', f ", f '", f(4), f(5),... Example10: Find the fifth derivativeoff ...
5. 6. 7. 8. 9. 10. Newton’s Law of UniversalGravitationstatesthat the gravitationalforcebetweentwo masses(say, the earthand the ...
4. 5. 6. 7. 8. 9. 10. 11. 12. 13. Derivativesof TrigonometricFunctions LearningObjectives A studentwill be able to: Computethe ...
The derivativebecomes Therefore, It will be left as an exerciseto provethat The derivativesof the remainingtrigonometricfunction ...
Solution: It is possibleto provethis relationby the definitionof the derivative.However, we use a simplermethod. Since then Exam ...
We finallyget the slopeto be approximately Example4: Ify= secx, findy" (π/3). Solution: Substitutingforx= π/3, Thusy" (π/3)= 14. ...
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