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Student Name Name of the Professor Date Problem Solving On Integral Calculus Q4) By differentiating the expression for charge q=255cos1000t µC. show that the current is i=-255sin1000t mA.Answer:- In mathematics differential calculus is a subfield of calculus which is mainly concerned with the study of rates at which quantities change. Mainly there are two parts in calculus. 1. Differential calculus 2. Integral calculus The primary objects of study in differential calculus are the derivative of a function related notions such as differential and their applications. The derivative of a function at a chosen input value describes the rate of change of the function near that input value. The process of finding a derivative is called differentiation. Geometrically the derivative at a point is the slope of the tangent line to the graph of the function at that point provided that derivative exists and is defined at that point. This by applying Laplace transform is the equation no (1) we will get: - [S2X(S)-SX(0-)-X՛(0-)]+ 6.25 [SX(S)-X(0-)]+25.8X(S)=0 Here initial condition is X(0-)=0.01m [given] [S2X(S)-(0.01)SX(S)]+6.25[SX(S)-0.01]+25.8X(S)=0 S2X(S)+6.24SX(S)+25.8X(S)-0.0625=0 S2X(S)+6.24SX(S)+25.8X(S)=0.0625 X(S)= 0.0625/(S2+6.24S+25.8) By solving the quadratic equation (S2+6.24S+25.8) we can find S= [ -6.24+-√{(6.25)2-25.8*4}]/2 S= -3.12+- 4.004j So X(S)=0.0625/{ (S+3.12-4.004j) (S+3.12+4.004j)} X(S)/0.0625= 1/{(S+3.12-4.004j) (S+3.12+4.004j)} X(S)/0.0625=A/(S+3.12-4.004j) + B/(S+3.12+4.004j) Now we can write for solving the equation A/(S+3.12-4.004j) + B/(S+3.12+4.004j) 1=A(S+3.12+4.004j)+B(S+3.12-4.004j) Putting the values of S=(-3.12-4.004j) and S=(-3.12+4.004j) we will get A= 1/8.008j and B= -1/8.008j Now X(S)= 0.625/{8.8008j(S+3.12-4.004j)} - 0.625/{8.008j(S+3.12+4.004j)} = -0.078j * 1/{s+(3.12-4.004j)} + 0.078j* 1/{ s+(3.12+4.004j)} By Applying Laplace inverse we will get:- X(t)= -0.078j e-(3.12-j4.004) u(t)+ 0.078je-(3.12+j4.004)u(t) (Ans) References Series resonance circuit Retrieved from: www.electronics-tutorials.ws IVP’s with Laplace Transforms. Retrieved from: tutorial.math.lamar.edu Equation for a spring-mass system. Retrieved from: iitg.vlab.co.in of differential equation. Retrieved from: en.wikipedia.org Theories of integration and differentiation. Retrieved from: www.icaiknowledgegateway.org differential equation of RLC circuit. Retrieved from: http://www.math.ubc.ca/~feldman/m121/RLC.pdf [...]

## Order Description:

I have to do 3 questions in total. One of the questions is relative to each other so in total, question 4, 5 and 6 needs to be attempt

Subject Area: Mathematics

Document Type: Dissertation Proposal