1019 WordsApr 13, 20155 Pages

MEI Core 4
Parametric Equations
Chapter Assessment
1. A curve is defined by the parametric equations x = 2t 2 , y = 4t . (i) By eliminating the parameter, find the cartesian equation of the curve. (ii) Find the equation of the tangent to the curve at the point A with parameter t = 2. (iii) Show that the tangent does not meets the curve again. (iv) The normal of the curve at A cuts the curve again at B. Find the coordinates of B. [3] [4] [3] [5]
2. Find the turning points of the curve with parametric equations x = 3t , y = 12t − t 3 and distinguish between them. [5] 3. A circle is defined by the parametric equations x = 1 + 2 cos θ , y = 3 + 2sin θ . (i) Sketch the circle. dy (ii) Find at the point with parameter θ . dx (iii) Find the equation of the tangent at the point with parameter θ . π (iv) Find the coordinates of the point where θ = . 3 π (v) Find the equation of the normal at the point where θ = . 3
[2] [3] [3] [2] [4]
4. A line is defined by the parametric equations x = cos 2t , y = sin 2 t dy . (i) Find dx (ii) Find the cartesian equation of the line.
[3] [3]
Total 40 marks
© MEI, 01/09/08
1/5
MEI C4 Parametric Equations Assessment solutions
Solutions to Chapter Assessment
1. (i)
x = 2t 2 y = 4t ⇒ t =
Insert into
y
4
:
2 y2 ⎛y⎞ x = 2⎜ ⎟ ⇒ x = 8 ⎝4⎠ 2 ⇒ y = 8x
(ii) x = 2t 2 ⇒
dx = 4t dt dy y = 4t ⇒ =4 dt dy dy / dt 4 1 = = = dx dx / dt 4t t dy 1 = When t = 2: dx 2
[3]
When t = 2: x = 2 × 2 2 = 8 and y = 4 × 2 = 8
1 So the tangent has gradient 2 and passes through (8, 8) 1 Equation of tangent is y − 8 = 2 ( x − 8 ) 2 y − 16 = x − 8 2y = x + 8
(iii) Where tangent meets curve, 2 × 4t = 2t + 8
2
[4]
2t 2 − 8t + 8 = 0
t 2 − 4t + 4 = 0
(t − 2)2 = 0 The only root is the repeated root at t = 2 (the point of the tangent) so the tangent doesn’t meet the curve again. [3] 1 (iv) Gradient of tangent =

Related

## Equation Essay

1621 Words | 7 Pagescalculation coming up, so I think I can forgive myself for getting the units straightened out at this point. Notice that I've kept an extra significant digit in the result, though, just for safety's sake.) Now, if we continue working from the last equation above Example 4, we can cross-multiply and solve for mu: [pic] Is this correct? Well, it is a reasonable answer, and notice that the units work out correctly, which is always a good, quick check. [pic] Case 2: v < videal: A free-body

## Accounting Equation Essay

379 Words | 2 PagesAccounting Equation The foundation for the double-entry bookkeeping system begins with the accounting equation. The total debits equal the total credits for each transaction. The accounting equation gives us a simple way to understand how these three amounts relate to each other. The accounting equation for sole proprietorship is:Assets = Liabilities + Owner’s EquityThe accounting equation for a corporation is:Assets = Liabilities + Stockholders’ Equity The accounting equation is basis for most

## Balancing Equation Essay

1591 Words | 7 Pageschemical reaction. Product is the substance that was created when the reactant underwent chemical reactions. A chemical equation is an expression that shows the reactants involved and products that were created in a chemical reactyion. A chemical reaction is typically in the format: I. Formulas show chemistry at a standstill. Equations show chemistry in action. A. Equations show: 1. the reactants which enter into a reaction. 2. the products which are formed by the reaction. 3. the

## Drake Equation Essay

2270 Words | 10 PagesCommentary on the Drake Equation Date: June 08, 2011 For almost a century while science has been making headway on many a front mankind has been plagued with one question ----- are we alone in this universe, or are there sentient beings on other planets and stars too? Contemporary proliferation of materials in the news media, the publishing world and cinematography on this topic is testimony to the fact that this is a question that raises eyebrows and attracts our attention. Pro-active endeavors

## Equations of Motion Essay

449 Words | 2 PagesName: Period: Equations of Motion 2A: a = ∆v/t Show all work and include units 1. The velocity of a car changes from 60 meters per second north to 45 meters per second north in 5.0 seconds. The magnitude of the car's acceleration is 2. A bicyclist accelerates from rest to a speed of 5.0 meters per second in 10 seconds. During the same 10 seconds, a car accelerates from a speed of 22 meters per second to a speed of 27 meters per second. Compared to the acceleration of the bicycle

## Accounting Equation Essay

452 Words | 2 PagesAccounting Equation ACC/300 10 September 2013 Accounting Equation Assets, liabilities and owners equity are three things that measure a company’s financial health (Investopedia). In order for a company to succeed financial statements, balance sheets, income statements and statements of retained earnings have to be accurate. This paper will describe the accounting equation as it relates to components of the balance sheet. The accounting equation is the most common among the various balance

## Quadratic Equations Essay

1109 Words | 5 PagesEmma Hancock Honors Algebra II 13 March 2014 Quadratic Equations: Assignment One 1. Find two consecutive whole numbers such that the sum of their squares is 265. 1st number: n 2nd number: n + 1 n2 + ( n + 1 )2 = 265 n2 + n2 + 2n + 1 = 265 2n2 + 2n - 264 = 0 2 ( n2 + n - 132 ) = 0 ( n2 + n - 132 ) / 2 = 0 / 2 n2 + n - 132 = 0 ( n - 11 ) ( n + 12) = 0 n - 11 = 0 n - 11 + 11 = 0 - 11 n = 11 n + 12 = 0 n + 12 - 12 = 0 - 12 n = -12 ( -12 is not a whole number so n must be positive 11) n + 1 = 11 + 11

## Quadratic Equation Essay

1417 Words | 6 PagesQuadratic Equation given the roots or given the sum and product of roots To find a quadratic equation that has given roots, we used reverse factoring and reverse completing the square. STEPS 1. Given the roots, set them equal to x. Example: (4, -2) x =4 or x = − 2 2. Make each equation equal zero x − 4= 0 or x + 2= 0 These expressions are the factors 3. Using FOIL (x − 4)(x + 2)= 0 4. Combine like terms x² +2x − 4x − 8 5. So the quadratic equation will

## Cfd on Equation Essay

421 Words | 2 Pagesapproach than two-equation or second-order closure models, one-equation models have been somewhat unpopular and have not showed a great deal of success. One notable exception was the model formulated by Bradshaw, Ferris, and Atwell (1967), whose model was tested against the best experimental data of the day at the 1968 Stanford Conference on Computation and Turbulent Boundary Layers. There has been some renewed interest in the last several years due to the ease with which one-equation models can be

## Quadratic Equations Essay

312 Words | 2 PagesQuadratic Equations Quadratic Equations A quadratic equation is used for various mathematical functions. “An equation is called a quadratic equation or a second-degree equation when the largest exponent of the variable is two” (Bluman, 2012). The following problems show a method from India to solve quadratic equations. An interesting method for solving quadratic equations came from India. The steps are: a) Move the constant term to the right side of the equation. b) Multiply each

### Equation Essay

1621 Words | 7 Pages### Accounting Equation Essay

379 Words | 2 Pages### Balancing Equation Essay

1591 Words | 7 Pages### Drake Equation Essay

2270 Words | 10 Pages### Equations of Motion Essay

449 Words | 2 Pages### Accounting Equation Essay

452 Words | 2 Pages### Quadratic Equations Essay

1109 Words | 5 Pages### Quadratic Equation Essay

1417 Words | 6 Pages### Cfd on Equation Essay

421 Words | 2 Pages### Quadratic Equations Essay

312 Words | 2 Pages

- Kavita Essay
- Culture Differences and English Teaching Essay
- Oliver Fitzherbert Essay
- Cloud Computing Essay
- How Are Steinbeck’s Views on the Inequalities in Society Shown in of Mice and Men? Essay
- With Reference to Examples Suggest How Successful Partnership Schemes Between Government and Private Sector Have Been (10 Marks) Essay