Question

Calculus

(a) The charge q(t) stored on a capacitor having capacitance C (measured in Farads),discharging through a resistor of resistance R (measured in Ohms), is given by

q(t)=Q e^{-\frac{t}{R C}}

where Q is the initial charge (measured in Coulombs), andt is time (measured in seconds). Suppose Q = 50 and C= 0.25.

(1) For R = 2.5, calculate the charge in the capacitor after 2 seconds have elapsed.

Suppose that for a different resistor we are given that 40% of the initial charge isost in the first 3 seconds. Find the value of R.(4 marks)

A voltage of the form v = 8 sin(2t) is applied to an electrical circuit. The steady-state output current i flowing through the circuit is given by the sinusoidal oscillation

i=2 \sin (2 t)+5 \cos (2 t)

The voltage is measured in volts and the current is measured in amps.

State the angular frequency of oscillation in radians per second and the fundamental period of oscillation in seconds of the input voltage (that is the minimum time required for a single cycle of oscillation).(2 marks)

) Express the current i in the form i =R sin(wt+a) where a is in radians hence state the amplitude of the output steady state current

Verified

Question 45629

Calculus

III. The following Theorem was discussed in thesection about infinite series

Question 45628

Calculus

II. The following figure shows an infinite zigzag path with each zag occurs at an angle of t/4. Find the total length of this infinite zigzag path.

Question 45627

Calculus

I. Find the total area of the infinitely many triangles shown in the figure below.

Question 45477

Calculus

The bandwidth of a video being delivered on the internet has a continuous distributionon the interval (0,4), with a probability density function (PDF)
f_{X}(x)=\left\{\begin{array}{ll} \frac{c}{(1+x)^{2}} & \text { for } 0<x<4 \\ 0 & \text { otherwise. } \end{array}\right.
What is the probability that the bandwidth of a video is less than 0.6.

Question 45476

Calculus

You are given that P(A) = 0.5 and P(A U B) = 0.7. Student 1 assumes that A and B are independent and calculates P(B) based on that assumption. Student 2 assumes that A and B are mutually exclusive and calculates P(B) based on that assumption.Find the absolute difference between the two calculations.

Question 45456

Calculus

\text { Sketch a (relatively decent) contour diagram for } f(x, y)=\frac{y+1}{x-y} \text { which includes the }
c= 0, 1, 2, –2 level curves. (Pick a reasonable scale for your axes.)

Question 45455

Calculus

1. (3) Is there a function of two variables whose z = 0 level curve consists exactly of the circles x2 + y2 = 2 and x2 + y2= 9? If so, what is an example? If not, why not?

Question 45454

Calculus

\text { a) Consider the sphere } S:(x-2)^{2}+(y+3)^{2}+(z-1)^{2}=16
(b) Let P be the horizontal plane which passes through (4, 5, 2). Find the equation ofthe circle where P intersects S.
(a) Find anequation for the circle where S intersects the xz-plane.
(c) Find the point(s) where S intersects the x-axis.

Question 45453

Calculus

2. (5) Consider the (somewhat strange) function V (w) which returns the number of uniquevowels (a, e, i, o, u only) in the English word w.
(a) What is V(calculus)? V (brachiosaurus)?
(b) Describe (in words) the zeros of the function V, and giveexample.one
(c) What is the range of V?

Question 45452

Calculus

\text { Consider the function } f(x, y)=x-y^{2}+3 \text {. Sketch the cross-section of } f(x, y) \text { with }
\text { (a) } x=2
\text { (b) } y=-1

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