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 \(n^2+2\).
(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!
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?
(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: \(x^2+y^2=20\)
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.
(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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