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Topic | Concept | Subconcept 1 | Subconcept 2 | Subconcept 3 | Subconcept 4 | Subconcept 5 |
---|---|---|---|---|---|---|

Limits | Limit of a function and sequence | Order of approximation | Indeterminate form | Continuous function | Differential calculus | Derivative |

Inverse functions and differentiation | Integral calculus | Integration by parts | Differentiation under the integral sign | Solid of revolution | Antiderivative/Indefinite integral | Special functions and numbers |

Natural logarithm | Stirling's approximation | Hyperbolic function | Bernoulli numbers | Absolute numerical | Rectangle method | Simpson's rule |

Newton–Cotes formulas | Gaussian quadrature | Multivariable | Partial derivative | Disk integration | Curvature | Divergence theorem |

Series | Infinite series | Maclaurin series, Taylor series | Fourier series | Euler–Maclaurin formula |

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**Q1:**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.)See Answer**Q2:**If a and b are any two rational numbers, does each of the following expressions always represent a rational number? 2. а — b.See Answer**Q3:**Apply concepts from this unit of study to answer the question: Is it possible to combinef(x)=Vx with a second function g(x) to get a new function whose domain is all real numbers?See Answer**Q4:**Show the equation of the translation image. [DOK 2: 2 marks]See Answer**Q5:**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] 5See Answer**Q6:**Compare the relationship between the domains and ranges of a function and its inverse?Explain.See Answer**Q7:**3.) The base of a solid is formed by the region enclosed by x=.25 y, x = y whose cross sections perpendicular to the y axis form isosceles right triangles with the hypotenuse on the base. Determine the volume of the solid.See Answer**Q8:**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? See Answer**Q9:**We will find the volume of the solid generated by revolving the region R about z-axis where Ris the region bounded by y= 4 Vz and y= 5.zO Which of the following is the correct integral to find the volume by dishwasher method 2 \pi \int_{4}^{5} y\left(\frac{y}{b}-\frac{y^{2}}{b^{2}}\right) d y O=\int_{0}^{0.64}(4 \sqrt{x})^{2}-(5 x)^{2} d x \int_{0}^{0.64} \frac{y^{2}}{4^{2}}-\left(\frac{y}{5}\right)^{2} d y \left.\int_{0}^{0.64}(5 x)^{2}-4 \sqrt{x}\right)^{2} d x O Which of the following is the correct integral to find the volume by Cylindrical Shell method \pi \int_{0}^{0.64}(4 \sqrt{x})^{2}-(5 x)^{2} d x \left.\int_{0}^{0.64}(5 x)^{2}-4 \sqrt{2}\right)^{2} d x 2 \pi \int_{0}^{3.2} y\left(\frac{y}{5}-\frac{y^{2}}{5^{2}}\right) d y 2 \pi \int_{0}^{0.64} y\left(\frac{y}{5}-\frac{y^{2}}{4^{2}}\right) d ySee Answer**Q10:**Students were generally more successful with this question than with question 8. The main big issues were that some students either did not apply the chain rule at all, while others applied the product rule instead, using In() as one of the parts. A smaller but common pitfall was not simplifying the equation at the end to -tan(x).See Answer**Q11:**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.See Answer**Q12:**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]See Answer**Q13:**4) The figure shows a concrete column that has an elliptical cross section. Find the volume of concrete in the column.See Answer**Q14:**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]See Answer**Q15:**2. Suppose that -a? < f(x) < x? for all x. Use this to find out what possiblevalues we could have for lim f(r) as done previously. Make sure to draw thenumberline. 3. Suppose that -r + 62? – 12x + 6 < f(x) <x' - 8x + 24x? – 32.x + 19 forall x. Use this to find out what possible values we could have for lim f(r) asdone previously. Make sure to draw the numberline.See Answer**Q16:**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)See Answer**Q17:**Find the missing coordinate of P, using the fact that P lies on the unit circle in the given quadrant. See Answer**Q18:**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.) See Answer**Q19:**Locate the position of any stationary points of the function f(x, y)=8 y^{3}+x^{2}-23 y Enter your answer in the form (x, y). Enter the point with the highest y coordinate first, then the lowest. Type N for both points if there are no stationary points. The position of the stationary point with the highest y coordinate is given by The position of the stationary point with the lowest y coordinate is given by See Answer**Q20:**To graph the field produced by F at any point in the coordinate plane.Take a picture to show your work. See Answer

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