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Calculus, MET1 2023 VCE SM-Bank 5

Let  \(f: R \rightarrow R\),  where  \(f(x)=2-x^2\).

  1. Calculate the average rate of change of \(f\) between \(x=-1\) and \(x=1\).  (1 mark)

  2. Calculate the average value of \(f\) between \(x=-1\) and \(x=1\).  (2 marks)

  3. Four trapeziums of equal width are used to approximate the area between the functions  \(f(x)=2-x^2\)  and the \(x\)-axis from \(x=-1\) to \(x=1\).
  4. The heights of the left and right edges of each trapezium are the values of \(y=f(x)\), as shown in the graph below.

  1. Find the total area of the four trapeziums.  (2 marks)

Show Answers Only

a.    \(0\)

b.    \(\dfrac{5}{3}\)

c.    \(\dfrac{13}{4}\)

Show Worked Solution
a.     \(\text{Average rate of change}\) \(=\dfrac{f(1)-f(-1)}{1-(-1)}\)
    \(\dfrac{1-1}{2}\)
    \(=0\)

 

b.    \(\text{Avg value}\) \(=\dfrac{1}{1-(-1)}\displaystyle\int_{-1}^{1} \Big(2-x^2\Big)\,dx\)
    \(=\dfrac{1}{2}\displaystyle\Bigg[2x-\dfrac{x^3}{3}\displaystyle\Bigg]_{-1}^{1}\)
    \(=\dfrac{1}{2}\displaystyle\Bigg[\Bigg(2.(1)-\dfrac{(1)^3}{3}\Bigg)-\Bigg(2.(-1)-\dfrac{(-1)^3}{3}\Bigg)\Bigg]\)
    \(=\dfrac{1}{2}\times\dfrac{10}{3}=\dfrac{5}{3}\)

 

c.     \(\text{Total Area}\) \(=2\times\ \text{Area from 0 to 1}\)
    \(=2\times \dfrac{h}{2}\Big(f(0)+2.f(0.5)+f(1)\Big)\quad \text{where }h=\dfrac{1}{2}\)
    \(=\dfrac{1}{2}\Big(2+2\times\dfrac{7}{4}+1\Big)=\dfrac{13}{4}\)

Calculus, MET2 2022 VCAA 17 MC

A function `g` is continuous on the domain `x \in[a, b]` and has the following properties:

  • The average rate of change of `g` between `x=a` and `x=b` is positive.
  • The instantaneous rate of change of `g` at `x=\frac{a+b}{2}` is negative.

Therefore, on the interval `x \in[a, b]`, the function must be

  1. many-to-one.
  2. one-to-many.
  3. one-to-one.
  4. strictly decreasing.
  5. strictly increasing.
Show Answers Only

`A`

Show Worked Solution

The first property shows that the point (`a` , __ ) must be below (`b` , __ ) to give a positive rate of change.

The second property gives the midpoint of `a` and `b`. But this point has a negative instantaneous rate of change.

Therefore, this shows that the function `g(x)` has turning points and is in the shape of a cubic, so many-to-one is the correct description.

 
`=>A`


♦♦ Mean mark 39%.

Calculus, MET2 2021 VCAA 13 MC

The value of an investment, in dollars, after `n` months can be modelled by the function

`f(n) =  2500 xx (1.004)^n`

where  `n ∈ {0, 1, 2, ...}`.

The average rate of change of the value of the investment over the first 12 months is closest to

  1. $10.00 per month.
  2. $10.20 per month.
  3. $10.50 per month.
  4. $125.00 per month.
  5. $127.00 per month.
Show Answers Only

`B`

Show Worked Solution

`text(By CAS):`

`text{Average ROC}` `={f(12) – f(0)}/{12 – 0}`
  `= 10.2229 …`

`=> B`

Calculus, MET2 2019 VCAA 3 MC

Let  `f: R\ text(\){4} -> R, \ f(x) = a/(x - 4),\ \ text(where)\ \ a > 0`.

The average rate of change of  `f`  from  `x = 6`  to  `x = 8`  is

A.   `a log_e(2)`

B.   `a/2 log_e(2)`

C.   `2a`

D.   `-a/4`

E.   `-a/8`

Show Answers Only

`E`

Show Worked Solution

`f(6) = a/2`

`f(8) = a/4`

`:.\ text(Average ROC)` `= (a/4 – a/2)/(8 – 6)`
  `= -a/8`

`=>   E`

Calculus, MET2 2017 VCAA 9 MC

The average rate of change of the function with the rule  `f(x) = x^2 - 2x`  over the interval  `[1, a]`, where  `a > 1`, is `8`.

The value of `a` is

  1.  `9`
  2.  `8`
  3.  `7`
  4.  `4`
  5.  `1+ sqrt2`
Show Answers Only

`A`

Show Worked Solution
`(f(a) – f(1))/(a – 1)` `= ((a^2-2a)-(1-2))/(a-1)`
`8` `=(a-1)^2/(a-1),\ \ \ (a>0)`
`8` `=a-1`
`a` `=9`

`=> A`

Calculus, MET2 2007 VCAA 4 MC

The average rate of change of the function with rule  `f(x) = x^3 - sqrt (x + 1)`  between  `x = 0`  and  `x = 3`  is

A.   `0`

B.   `12`

C.   `26/3`

D.   `25/3`

E.   `8`

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`C`

Show Worked Solution

`text(Define)\ \ f(x) = x^3 – sqrt (x + 1)\ \ text(on CAS)`

`text(Average ROC)` `= (f(3) – f(0))/(3 – 0)`
  `= 26/3`

`=>   C`

Calculus, MET2 2010 VCAA 2 MC

For  `f(x) = x^3 + 2x`, the average  rate of change with respect to `x` for the interval  `[1, 5]`  is

A.   `18`

B.   `20.5`

C.   `24`

D.   `32.5`

E.   `33`

Show Answers Only

`E`

Show Worked Solution
`text(Average ROC)` `= (f(5) – f(1))/(5 – 1)`
  `=(135-3)/4`
  `= 33`

`=>   E`

Calculus, MET2 2016 VCAA 4 MC

The average rate of change of the function  `f` with rule  `f(x) = 3x^2 - 2 sqrt(x + 1)`, between  `x = 0 and x = 3`, is

A.   `8`

B.   `25`

C.   `53/9`

D.   `25/3`

E.   `13/9`

Show Answers Only

`D`

Show Worked Solution
`text(Average ROC)` `= {f(3) – f(0)}/(3 – 0)`
  `=[(27-2xxsqrt4) – (-2)]/(3-0)`
  `= 25/3`

`=>   D`

Algebra, MET2 2012 VCAA 2 MC

For the function with rule  `f(x) = x^3 - 4x`, the average rate of change of  `f(x)`  with respect to `x` on the interval `[1,3]` is

A.   `1`

B.   `3`

C.   `5`

D.   `6`

E.   `9`

Show Answers Only

`=> E`

Show Worked Solution
`text(Average ROC)` `= (f(3) – f(1))/(3 – 1)`
  `=(15-(-3))/2`
  `= 9`

`=> E`

Calculus, MET2 2013 VCAA 6 MC

For the function  `f(x) = sin (2 pi x) + 2x,`  the average rate of change for  `f(x)`  with respect to  `x`  over the interval  `[1/4, 5]`  is

A.   `0`

B.   `34/19`

C.   `7/2`

D.   `(2 pi + 10)/4`

E.   `23/4`

Show Answers Only

`B`

Show Worked Solution

`text(Average rate of change for)\ \f(x)\ \ text(over)\ \  [1/4, 5]`

`= (f(5) – f(1/4))/(5 – 1/4)`
`= (sin (10 pi) + 10 – (sin\ pi/2 +1/2))/(19/4)`
`= 34/19`

`=>   B`