One Variable Calculus with SageMath









Differentiating logarithm and exponential functions

differentiate ln x from first principles. • differentiate ex. Contents. 1. Introduction. 2. 2. Differentiation of a function f(x).
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EXERCISES IN MATHEMATICS Series F No. 2: Answers First

1. Differentiate from first principles y = x2 − 4x. Answer. We have y + δy = (x + δx)2 − 4(x 
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Untitled

log y. = = 0720 log (cos x x) (0. Lt sin x lim. →0 log y = 0. COS X Differentiation from First Principle (AB-Initio Method). Let f(x) is a function ...
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1 Theory of convex functions

1 mar. 2016 Let's first recall the definition of a convex function. ... In words this means that if we take any two points x
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CONTINUITY AND DIFFERENTIABILITY

The derivative of logx. w.r.t. x is. 1 x. ; i.e.. 1. (log ) d x dx x. = . 5.1.12 Logarithmic differentiation is a powerful technique to differentiate 
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Recapitulation of Mathematics

is known as the first principle of differential calculus. 1.2 Differential Coefficient of a Function at a Point. The value of the derivative of f(x) 
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One Variable Calculus with SageMath

13 juil. 2019 Find the derivative of h(x) = log(x) + x5 + sin(x) using the first principle. sage: ax=var('a
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Week 3 Quiz: Differential Calculus: The Derivative and Rules of

is important to note the this function is undefined at x = 3. Answer: Note first that for any real number t we have −1 ≤ sint ≤ 1 so −1 ≤ sin(1 x. ) ...
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Course Contents: Topic and Contents Hours Marks : 15SC02M

Derivatives of functions ofx sin x
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Differentiating logarithm and exponential functions

differentiate ln x from first principles. • differentiate ex. Contents. 1. Introduction. 2. 2. Differentiation of a function f(x).
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213450 One Variable Calculus with SageMath

One Variable Calculus with SageMath

Ajit Kumar

Institute of Chemical Technology, Mumbai

ajit72@gmail.com

1 Limit

Example1.Find the limx!0xjxj.

sage :x =var( x sage :f (x)=x/ abs (x) sage :p = p lot(f,-2,2,figsize=5)-2-112 -1-0.50.51sage:f .limit(x=0,dir='+') x |--> 1 sage :f .limit(x=0, dir x |--> -1 sage :f .limit(x=0) x |--> und

Example2.Leta2R. Find limx!1(1 +a=x)x.

sage :a =var( a sage :f = ( 1+a/x)^x sage :f .limit(x=infinity) e^a Example3.Explore the limit ofg(x) =xsin(1=x) atx= 0. sage :g (x)=x*sin(1/x) sage :p = p lot(g,-0.1,0.1,color= red ,figsize=5)

Distributed during \Teachers Enrichment Course in Undergraduate Mathematics Curriculum" at IIT Guwahati

during July 1-13, 2019 1 -0.1-0.050.050.1 -0.08-0.06-0.04-0.020.020.040.06sage:g .limit(x=0,dir='-') x |--> 0 sage :g .limit(x=0, dir x |--> 0 sage :g .limit(x=0) x |--> 0

Exercise1.Compute the following limits:

1. lim x!0tanxxx 3. 2. lim x!0(1 + sinx)cot2x. 3. lim x!1(1 +a=x)x. 4. If pdollars is compoundedntimes per year at an annual interest rate ofr, the money will be worthp(1 +r=n)ntdollars after t years. How much will the money be worth after t years if it is compounded continuously? (n! 1)

1.1 Piecewise Dened Functions

sage :f 1(x)= x *sin(1/x) sage :f 2(x)= 1 -x^2 sage :f 3(x)=x*cos(1/x) sage :f = p iecewise([[(-2,0),f1],[(0,2),f2],[(2,3),f3]]) sage :p =plot(f,(x,-2,3),figsize=5)-2-1123 -3-2-112

1.2 Limit of Sequences

Example4.limn!11 +1n

n: sage :n =var( n 2 sage:f (n)=(1+1/n)^n sage :l imit(f(n),n=oo) e

Example5.Assumea >0. Find limn!1a1=n.

sage :v ar( a a sage

One Variable Calculus with SageMath

Ajit Kumar

Institute of Chemical Technology, Mumbai

ajit72@gmail.com

1 Limit

Example1.Find the limx!0xjxj.

sage :x =var( x sage :f (x)=x/ abs (x) sage :p = p lot(f,-2,2,figsize=5)-2-112 -1-0.50.51sage:f .limit(x=0,dir='+') x |--> 1 sage :f .limit(x=0, dir x |--> -1 sage :f .limit(x=0) x |--> und

Example2.Leta2R. Find limx!1(1 +a=x)x.

sage :a =var( a sage :f = ( 1+a/x)^x sage :f .limit(x=infinity) e^a Example3.Explore the limit ofg(x) =xsin(1=x) atx= 0. sage :g (x)=x*sin(1/x) sage :p = p lot(g,-0.1,0.1,color= red ,figsize=5)

Distributed during \Teachers Enrichment Course in Undergraduate Mathematics Curriculum" at IIT Guwahati

during July 1-13, 2019 1 -0.1-0.050.050.1 -0.08-0.06-0.04-0.020.020.040.06sage:g .limit(x=0,dir='-') x |--> 0 sage :g .limit(x=0, dir x |--> 0 sage :g .limit(x=0) x |--> 0

Exercise1.Compute the following limits:

1. lim x!0tanxxx 3. 2. lim x!0(1 + sinx)cot2x. 3. lim x!1(1 +a=x)x. 4. If pdollars is compoundedntimes per year at an annual interest rate ofr, the money will be worthp(1 +r=n)ntdollars after t years. How much will the money be worth after t years if it is compounded continuously? (n! 1)

1.1 Piecewise Dened Functions

sage :f 1(x)= x *sin(1/x) sage :f 2(x)= 1 -x^2 sage :f 3(x)=x*cos(1/x) sage :f = p iecewise([[(-2,0),f1],[(0,2),f2],[(2,3),f3]]) sage :p =plot(f,(x,-2,3),figsize=5)-2-1123 -3-2-112

1.2 Limit of Sequences

Example4.limn!11 +1n

n: sage :n =var( n 2 sage:f (n)=(1+1/n)^n sage :l imit(f(n),n=oo) e

Example5.Assumea >0. Find limn!1a1=n.

sage :v ar( a a sage
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  6. derivative of log sec x by first principle
  7. derivative of log cos inverse x by first principle
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