Grind #1 - Arithmetic Sequences & Series

Consider the following sequence:

\begin{align}7,11,15,19…\end{align}

**(a)** Show that this sequence is arithmetic.

**(b)** What is the next term in this sequence?

**(c)** What is the general term for this sequence?

**(d)** What is the \(100\)th term in this sequence?

The general term of a sequence is given by:

\begin{align}T_n=4n+1\end{align}

**(e)** What are the first three terms of this sequence?

**(f)** What is \(30\)th term of this sequence?

**(g)** Which term of the sequence is \(61\)?

The general term \(T_n\) of a particular sequence is given by

\begin{align}T_n = 2n+1\end{align}

**(h)** Show that that the sum of the first \(n\) terms of this sequence is \(n^2+2\).

Answers

**(a)** There is a common difference between successive terms of \(4\).

**(b)** \(23\)

**(c)** \(T_n=4n+3\)

**(d)** \(403\)

**(e)** \(5,9,13\)

**(f)** \(121\)

**(g)** \(15\)th

**(h)** The answer is already in the question!

Grind #2 - Coordinate Geometry of the Circle

Consider the points \((1,2)\), \((3,-4)\) and \((5,-6)\).

**(a)** Find the equation of the circle whose circumference passes through these three points.

Consider the circle \(x^2+y^2=10\).

**(b)** Show that \(y=-3x-10\) is a tangent to this circle.

A circle has a centre \((3,-1)\) and the point \((2,5)\) on its circumference.

**(c)** What is the equation of that circle?

Circle \(A\) has a centre \((-2,4)\) and the origin on its circumference.

Circle \(B\) has a centre at the origin and the point \((-2,4)\) on its circumference.

**(d)** What are the equations of both circles?

Answers

**(a) **\(x^2+y^2-22x-4y+25=0\)

**(b)** The answer is already in the question!

**(c)** \((x-3)^2+(y+1)^2=38\)

**(d) A:** \((x+2)^2+(y-4)^2=20\) **B:** \(x^2+y^2=20\)

Grind #3 - Turning Points

Consider the following function:

\begin{align}y(x)=3x^2+6x+4\end{align}

**(a)** Calculate \(\dfrac{dy}{dx}\)

**(b)** What is the turning point of \(y(x)\)?

Consider the following function:

\begin{align}y(x)=x^2-10x\end{align}

**(c)** What is the turning point of \(y(x)\)?

**(d)** By using the second derivative test, show that this turning point is a local *minimum*.

Consider the following function:

\begin{align}y(x)=x^3-3x^2-9x+6\end{align}

**(e)** What are the turning points of \(y(x)\)?

**(f)** By using the second derivative test, state whether each turning point is a local maximum or a local minimum.

Answers

**(a)** \(\dfrac{dy}{dx}=6x+6\)

**(b)** \((-1,1)\)

**(c)** \((5,-25)\)

**(d)** Minimum as the second derivative is positive.

**(e)** \((-1,11)\) and \((3,-21)\)

**(f)** \((-1,11)\) is a maximum as the second derivative is negative at that point. \((3,-21)\) is a minimum as the second derivative is positive at that point.

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