SmarterEd

Aussie Maths & Science Teachers: Save your time with SmarterEd

Calculus, 2ADV C3 2025 MET2 16 MC

Consider the function  \(h(x)=a\, \log _e(b x)\), where \(a, b \in R\).

Given that its derivative \(h^{\prime}(x)\) has range \((0, \infty)\), which of the following must be true?

  1. \( a>0\)  only
  2. \( a>0\)  and  \(b<0\)
  3. \(a>0\)  and  \(b>0\)
  4. \(a b>0\)
Show Answers Only

\(D\)

Show Worked Solution

\(h(x)=a\, \log _e(b x)\)

\(h^{\prime}(x)=a\, \times \dfrac{b}{bx}=\dfrac{a}{x}\)

\(\text{Consider} \ \ h(x)=a\, \log _e(b x) \ \Rightarrow \ b x>0\)
 

\(\text{Case 1:} \ \ b>0 \ \Rightarrow \ x>0 \ (\text{since} \ \ b x>0 )\)

\(\dfrac{a}{x} \ \ \text{is only positive when}\ \  a>0\)
 

\(\text {Case 2:} \ \ b<0 \ \Rightarrow \ x<0 \ \ (\text{since} \ \ b x>0)\)

\(\dfrac{a}{x} \ \ \text{is only positive when} \ \  a<0\)

\(\text{In both cases,} \ ab>0\)

\(\Rightarrow D\)

Calculus, 2ADV C3 2025 HSC 9 MC

The diagram shows the graph of  \(y=f^{\prime}(x)\).
 

Given  \(f(1)=6\), which interval includes the best estimate for \(f(1.1)\) ?

  1. \([6.2,6.4)\)
  2. \([6.0,6.2)\)
  3. \([5.8,6.0)\)
  4. \([5.6,5.8)\)
Show Answers Only

\(B\)

Show Worked Solution

\(\text{Gradient of \(f(x)\)  at  \(x=1\)  is 2 (see graph).}\)

\(\text{Gradient of \(f(x)\)  at  \(x=1.1\)  is slightly below 2 (see graph).}\)

\(\text{As \(x\) increases 0.1 (from 1.0 to 1.1), \(y\) will increase less than 0.2 units.}\)

\(\therefore f(1.1) \in [6.0,6.2)\)

\(\Rightarrow B\)

♦♦♦ Mean mark 27%.

Calculus, 2ADV C3 SM-Bank 2

Given  \(y=x e^{-3 x}\), prove that

\(\dfrac{d^2 y}{d x^2}+6 \dfrac{d y}{d x}+9 y=0\)   (3 marks)

Show Answers Only

\(\text{Proof (See worked solution}\)

Show Worked Solution

\(y=x e^{-3 x}\)

\(\dfrac{d y}{d x}=e^{-3 x}-3 x e^{-3 x}\)

\(\dfrac{d^2 y}{d x^2}\) \(=-3 e^{-3 x}-3 e^{-3 x}+3 \cdot 3 x e^{-3 x}\)
  \(=-6 e^{-3 x}+9 x e^{-3 x}\)

 
\(\text {Substituting into equation: }\)

\(\dfrac{d^2 y}{d x^2}+6 \dfrac{d y}{d x}+9 y\)

\(=-6 e^{-3 x}+9 x e^{-3 x}+6\left(e^{-3 x}-3 x e^{-3 x}\right)+9 x e^{-3 x}\)

\(=-6 e^{-3 x}+9 x e^{-3 x}+6 e^{-3 x}-18 x e^{-3 x}+9 x e^{-3 x}\)

\(=0\)

Calculus, 2ADV C3 2014 HSC 14a

Find the coordinates of the stationary point on the graph  `y = e^x − ex`, and determine its nature.   (3 marks)

Show Answers Only

`(1,0)\ =>text(MINIMUM)`

Show Worked Solution
`y` `= e^x – ex`
`dy/dx` `= e^x – e`
`(d^2 y)/(dx^2)` `= e^x`

 
`text(S.P. when)\ \ dy/dx = 0`

`e^x – e` `= 0`
`e^x` `= e^1`
`x` `= 1`

 
`text(At)\ \ x = 1`

`y` `= e^1 – e = 0`
`(d^2 y)/(dx^2)` `= e > 0\ \  => text(MIN)`

 
`:.\ text(MINIMUM S.P. at)\ (1,0)`

Calculus, 2ADV C3 2010 HSC 8d

Let  `f(x) = x^3-3x^2 + kx + 8`, where `k` is a constant.

Find the values of `k` for which `f(x)` is an increasing function.   (2 marks)

Show Answers Only

`k>3`

Show Worked Solution
`f(x)` `= x^3-3x^2 + kx + 8`
`f^{′}(x)` `= 3x^2-6x + k`

  
`f(x)\ text(is increasing when)\ \ f^{′}(x) > 0`

`=> 3x^2-6x + k > 0`

♦♦ Mean mark 28%.
MARKER’S COMMENT: The arithmetic required to solve `36-12k<0`  proved the undoing of many students.

 

`f^{′}(x)\ text(is always positive)`

`=> f^{′}(x)\ text(is a positive definite.)`

`text(i.e. when)\ \ a > 0\ text(and)\ Delta < 0`
 

`a=3>0`

`Delta = b^2-4ac`

`(-6)^2-(4 xx 3 xx k)` `<0`
`36-12k` `<0`
`12k` `>36`
`k` `>3`

 

`:.\ f(x)\ text(is increasing when)\ \ k > 3.`

Calculus, 2ADV C3 2013 HSC 12a

The cubic  `y = ax^3 + bx^2 + cx + d`  has a point of inflection at  `x = p`. 

Show that  `p= - b/(3a)`.   (2 marks)

Show Answers Only

 `text(Proof)\ \ text{(See Worked Solutions)}`

Show Worked Solution

`text(Show)\ \ p= – b/(3a)`

`y` `=ax^3 + bx^2 + cx + d`
`y prime` `=3ax^2 + 2bx + c`
`y″` `=6ax + 2b`

 
`text(Given P.I. occurs when)\ \ x = p`

`=> y″=0\ \ text(when)\ \ x=p`

`:.\ 6ap + 2b` `=0`
`6ap` `=-2b`
`p` `= -(2b)/(6a)`
  `=-b/(3a)\ \ \ text(… as required)`