Calculus Homework Help

I was attempting to draw a sinusoidal curve, y = sina, on some grid paper but didn't pay attention to where I was drawing! The curve I drew was translated away from the origin and was scaled in both x and y directions (but luckily not rotated). (a) The highest points, peaks, I draw was along y = 8 and the lowest points, troughs, along y = 2. I could make sense of the amplitude as how far the curve oscillated away from the middle (transversal) horizontal line of the curve. What is the amplitude? (Please turn over.)

If a and b are any two rational numbers, does each of the following expressions always represent a rational number? 2. а — b.

Show the equation of the translation image. [DOK 2: 2 marks]

Create a function f(x) with 4 different transformations. Then graph f(x) and its parent functiong(x). Label each of your graphs. [DOK 4: 5 marks] 5

4. Three people are watching a hot air balloon travel over their town. At a certain point in time,one person stands directly below the balloon, and the others look at it at certain angles. In the following image, A, B, and Care people, and D is the balloon. Person C is 384m directly below the balloon, Person B is 200m away from Person C, and the angle between Person A, the balloon, and Person B is 33°. How far is Person A from the hot air balloon?

3. During his stay at a hotel, a man decides to take a warm bath. He fills the tub with 200 liters of water and adds200 grams of liquid soap to the tub. The man mistakenly leaves the tap open, which keeps pouring water at arate of 1 liter/hour. Find the amount of soap y(t) left in the tub after time t. Hint: use liters, grams and hoursas units of measure, so you can just omit them and focus on the math problem! E.g., the initial amount of soapwill be 200 rather than 200g.

2) Find the area of the region enclosed between the two circles x² + y² = 1 and (x – 1)² + y² = 1. Draw a rough sketch as well.[5 marks]

3) Find the area of the region bounded by the parabola y = x² + 1 and the lines y = x,x = 0 and x = 2.Sketch the rough diagram as well.[5 marks]

Problem 6. (12 pts) Determine whether the following statement is true or false. If the statement is true, prove it, and, if it is false, provide a counterexample. \text { If } \sum_{n \in \mathbb{N}} a_{n} \text { is conditionally convergent, then } \sum_{n \in \mathbb{N}} n^{k} a_{n} \text { is divergent, for each } k \in(1,+\infty)

2. A particle P is fired from an origin O, which is at a distance L from the foot of a vertical cliff (AB)| height H. P is fired with speed U at an angle 0 to the horizontal and lands at a point C (see diagram You may neglect the effects of air resistance and assume that the gravitational field is g=-gk,here g is a constant and k is a unit vector (shown in the diagram). If the position of P relative to O is r(t), explain (in one sentence) why the equation of motion is f = -gk. Starting from the equation of motion above with r = x(t)i+ y(t)j, solve for z(t) and y(t). Hence obtain an expression for the time, T, at which the particle arrives at C in terms of U, 0, g and H;find the distance BC in terms of T, U, 0 and L. If the path of P passes through the point B, then show that the firing angle 0 can be determined from a quadratic equation for the quantity tan 6. (You are not required to solve the quadratic equation.)

Test the convergence of each of the following two series. \text { (i) } \sum_{n=1}^{\infty} \frac{n^{n}}{n !} \text { (ii) } \sum_{n=1}^{\infty}\left(\frac{2 n+3}{3 n+2}\right)^{n}

Let D:= {() e R² | |æl, \y] < 1,z101 +r < y + y° +y}, Please compute \iint_{D} 4 x^{3} y d x d y

\text { Show that the function } T(x)=\sum_{n=2}^{\infty}\left(\frac{1}{x-n \pi}+\frac{1}{n \pi}\right) \text { is well-defined } \text { on }[0, \pi] \text {. That is, prove that the domain of the function } T \text { contains the interval }[0, \pi] \text {. }

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Suppose a free motion mass-spring system with no damping is described by x^{11}+25 x=0, x(0)=-1, x^{\prime}(0)=10 Determine the position function and write it in the form x(t) = A cos (wt - a).

Consider an LRC-circuit with L= 0.5, R = 200, C = 0.001 and E(t) = 100 sin(60 t) a) Find the transient solution and determine whether the solution is overdamped, critically overdamped, or underdamped. (10 points) b) Find the steady periodic solution and write the full general solution. (10 points)

8. A banner has the shape of a right triangle with sides 6 feet, 8 feet and 10 feet and it weighs 12 lbs. (Note:the height of the triangle off the hypotenuse is 4.8 feet.) It is hung from the roof over the side of a building with the hypotenuse at the top. Set up the integral to compute the work required to lift the banner onto the roof of the building. Evaluate the integral to find the work.

2. A cylinder tank standing on its circular base is full of water, with the dimension given below. Write the integral needed to find the amount of work that would be needed to pump all of the water out of the tank to a point 3 feet above the tank? (Do not evaluate)а. b. How would your integral change if the tank is on its side? Write the integral to calculate the work need to pump all the water out of the tank to a point 3 feet above the tank, if the cylinder was laying on its side.(Do not have to evaluate the integral) a) h= 0 at the bottom b) Pick h=0 where the setup is the easiest.