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🔬 Tutorial problems beta

🔬 Tutorial problems beta#

Note

This problems are designed to help you practice the concepts covered in the lectures. Not all problems may be covered in the tutorial, those left are for additional practice.

\(\beta\).1#

Find the (Euclidean) distances between the following pairs of points.

  1. \((1, 3)\) and \((2, 4)\)

  2. \((-1, 2)\) and \((3, 3)\)

  3. \((\tfrac{3}{2},-2)\) and \((-5,1)\)

  4. \((x,y)\) and \((2x,y+3)\)

  5. \((a, b)\) and \((-a, b)\)

  6. \((a,3)\) and \((2+a,5)\)

[Sydsæter, Hammond, Strøm, and Carvajal, 2016] Exercises for Section 5.5, Question 1

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\(\beta\).2#

Suppose that the cost of producing \(Q\) units of a commodity is given by \(C(Q) = 1, 000 + 300Q + Q^2\).

  1. Compute \(C(0)\), \(C(100)\), and \(C(101) - C(100)\).

  2. Compute \(C(Q + 1) - C(Q)\) and explain the meaning of this expression.

[Sydsæter, Hammond, Strøm, and Carvajal, 2016] Exercises for Section 4.2, Question 6

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\(\beta\).3#

Consider the function \(f(x) = \frac{3x+6}{x-2}\).

  1. Find the domain of this function

  2. Show that \(5\) belongs to the range of this function. Hint: Find a value for \(x\) such that \(f(x) = 5\).

  3. Show that \(3\) does not belong to the range of this function.

[Sydsæter, Hammond, Strøm, and Carvajal, 2016] Exercises for Section 4.2, Question 14

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\(\beta\).4#

Suppose that the (Euclidean) distance between \((2,4)\) and \((5,y)\) is \(\sqrt{13}\).

  1. Find all possible values for \(y\).

  2. Explain geometrically why there must be exactly two distinct values for \(y\).

[Sydsæter, Hammond, Strøm, and Carvajal, 2016] Exercises for Section 5.5, Question 2

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\(\beta\).5#

To show that \(x^2+y^2-10x+14y+58 = 0\) is a circle, we argue like this: first, rearrange the equation to read \((x^2-10x) + (y^2 +14y) = -58\). Completing the two squares gives: \((x^2-10x+5^2) + (y^2 +14y+7^2) = -58 + 25 + 49\). This equation becomes \((x-5)^2 + (y+7)^2 = 16\), whose graph is a circle with center \((5,-7)\) and radius \(\sqrt{16} = 4\).

Use this method to find the center and the radius of the two circles with equations:

  • \(x^2+y^2+10x-6y+30 = 0\)

  • \(3x^2+3y^2+18x-24y = -39\)

[Sydsæter, Hammond, Strøm, and Carvajal, 2016] Exercises for Section 5.5, Question 5

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\(\beta\).6#

Find the domains of the following functions and then illustrate those domains in the \((x, y)\)–coordinate-plane.

  1. \(f(x,y)= \frac{x^2+y^3}{y-x+2}\)

  2. \(f(x,y)=\sqrt{2-(x^2 +y^2)}\)

  3. \(f(x,y)=\sqrt{(4-x^2 -y^2)(x^2 +y^2 -1)}\)

[Sydsæter, Hammond, Strøm, and Carvajal, 2016] Exercises for Section 11.1, Question 6

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