Calculus Answers

A capacitor circuit has been charged up to 12V and the instantaneous voltage is 𝒗 = 𝟏𝟐 (𝟏 − 𝒆 − 𝒕 𝜏)

The tasks are to: 

a) Draw a graph of voltage against time for 𝑣 = 12𝑉 and 𝜏 = 2𝑠, between 𝑡 = 0𝑠 and 𝑡 = 10𝑠.

b) Calculate the gradient at 𝑡 = 2𝑠 and 𝑡 = 4𝑠

c) Differentiate 𝒗 = 𝟏𝟐 (𝟏 − 𝒆 − 𝒕 𝜏) and calculate the value of 𝑑𝑣 𝑑𝑡 at 𝑡 = 2𝑠 and 𝑡 = 4𝑠.

d) Compare your answers for part b and part c. 

e) Calculate the second derivative of the instantaneous voltage ( 𝑑^2𝑣/𝑑𝑡^2 ). 

Use an appropriate technique to find the derivative of the following functions:

a. "[\nx\ny^3\/\n\n1\n+\nsec\n(\ny\n)\n]" = e^xy

^b. "(\u221a(\n3\nx\n^2\n+\n4)\/\n(\nx\n^2\n+\n1)^(1\/3^))(\u03c0^X)"

Solve the exact derivative of f(x) = x² at x = 5

1. Solve y = 5e^3x + sin x (dy/dx)
2. The velocity of a body as a function of time is given as ν (t) = 5e ^-2t  + 4, where t is in seconds, and v is in m/s. Solve the acceleration in m/s² at t = 0.6

find the area outside circle r = 1/2 and inside circle r = cosθ


In this assignment, you are asked to write a matlab program to solve the following open ended problem, and submit the program and written summary including methods employed, algorithm implementation, final results and recommendations on further improvement.

Background

A drug administered to a patient produces a concentration in the blood stream given by

                                               C(t)=Atexp(-t/3 )

Where c is in milligrams per millimeter, t is in hours and A is injected units. For a given drug the maximum safe concentration (in mg/ml) is denoted as c_max, and the critical concentration(in mg/ml), denoted as c_critical, is the concentration below which an additional amount of drug needs to be administered to the patient.

a.      Determine how the injected unit A affects the maximum safe concentration and when the maximum safe concentration occurs.

b.      Determine how the ratio between c_critical and c_max affects the duration between the first and the second injections.

Suggestions

if f(z) is an analytical function of z, then prove that ( d^2/dx^2 + d^2/ dy^2) log |f(z)|= 0

use chain rule to find dy/dx for given value of x

y=(u-1/u+1)^1/2, u=√x-1, for x=34/9