Mean value of function

5 The function f is defined by $$f ( x ) = 3 x ^ { 2 } \quad 1 \leq x \leq 5$$ 5

1. Find the mean value of f
   5
2. The function g is defined by $$\mathrm { g } ( x ) = \mathrm { f } ( x ) + c \quad 1 \leq x \leq 5$$ The mean value of \(g\) is 40
   Calculate the value of the constant \(c\)

1 Find the mean value of \(\mathrm { f } ( x ) = x ^ { 2 } + 6 x\) over the interval \([ 0,3 ]\).

9 In an electrical circuit, the alternating current \(I\) amps is given by \(\mathbf { I } =\) asinnt, where \(t\) is the time in seconds and \(a\) and \(n\) are positive constants. The RMS value of the current, in amps, is defined to be the square root of the mean value of \(I ^ { 2 }\) over one complete period of \(\frac { 2 \pi } { n }\) seconds. Show that the RMS value of the current is \(\frac { a } { \sqrt { 2 } }\) amps.

4 In this question you must show detailed reasoning.
Determine the mean value of \(\frac { 1 } { 1 + 4 x ^ { 2 } }\) between \(x = - 1\) and \(x = 1\). Give your answer to 3 significant
figures. figures.

2 Given that \(\mathrm { f } ( x ) = 3 x - 1\) find the mean value of \(\mathrm { f } ( x )\) over the interval \(4 \leq x \leq 8\) Circle your answer. 6111717

7 The function f is defined by $$f ( x ) = \frac { 1 } { \sqrt { x } } \quad 4 \leq x \leq 7$$ Find the mean value of f over the interval \(4 \leq x \leq 7\) Give your answer in exact form.

2 Find the mean value of \(3 x ^ { 2 }\) over the interval \(1 \leq x \leq 3\) Circle your answer.
[0pt] [1 mark] $$8 \frac { 2 } { 3 } \quad 10 \quad 13 \quad 26$$

3 The function \(\mathrm { f } ( x ) = x ^ { 2 } - 1\) Find the mean value of \(\mathrm { f } ( x )\) from \(x = - 0.5\) to \(x = 1.7\) Give your answer to three significant figures.
Circle your answer.

3 The function f is defined by $$f ( x ) = x ^ { 2 } \quad ( x \in \mathbb { R } )$$ Find the mean value of \(\mathrm { f } ( x )\) between \(x = 0\) and \(x = 2\) Circle your answer. \(\frac { 2 } { 3 }\) \(\frac { 4 } { 3 }\) \(\frac { 8 } { 3 }\) \(\frac { 16 } { 3 }\)

2
3 2 Find the mean value of the function \(\mathrm { f } ( x ) = 10 x ^ { 4 }\) between \(x = 0\) and \(x = a\) Circle your answer.
[0pt] [1 mark] \(10 a ^ { 3 }\) \(40 a ^ { 3 }\) \(2 a ^ { 4 }\) \(4 a ^ { 5 }\)