SmarterEd

Aussie Maths & Science Teachers: Save your time with SmarterEd

Calculus, 2ADV C3 SM-Bank 1 MC

The tangent to the graph of  `y = x^3 - ax^2 + 1`  at  `x = 1` passes through the origin.

The value of `a` is

  1. `1/2`
  2. `1`
  3. `3/2`
  4. `5/2`
Show Answers Only

`B`

Show Worked Solution
`y` `= x^3 – ax^2 + 1`
`dy/dx` `= 3x^2 – 2ax`

 
`text{At} \ \ x = 1 \ => \  y = 2-a, \ dy/dx = 3-2a`
 

`text{T} text{angent passes through} \ (1,2-a) \ text{and} \ (0,0)`

`m_text{tang} = 2 – a`

`text{Equating gradients:}`

`3-2a` `= 2-a`
`:. a` `= 1`

 
`=> B`

Calculus, 2ADV C3 2021 HSC 31

By considering the equation of the tangent to  `y = x^2 - 1`  at the point  `(a, a^2 - 1)`, find the equations of the two tangents to  `y = x^2 - 1`  which pass through `(3, –8)`.  (4 marks)

Show Answers Only

`y = 14x – 50`

`y = -2x – 2`

Show Worked Solution

`y = x^2 – 1`

♦♦ Mean mark 36%.

`(dy)/(dx) = 2x`

`text(At)\ \ x = a, (dy)/(dx) = 2a`
 

`text(Find equation of line)\ \ m = 2a, text(through)\ (a, a^2 – 1):`

`y – (a^2 – 1)` `= 2a(x – a)`
`y – a^2 + 1` `= 2ax – 2a^2`
`y` `= 2ax – a^2 – 1`

 
`text(If tangent passes through)\ (3, –8):`

`2a(3) – a^2 – 1` `= -8`
`6a – a^2 + 7` `= 0`
`a^2 – 6a – 7` `= 0`
`(a – 7)(a + 1)` `= 0`

 
`=> a = 7\ \ text(or)\ \ -1`
 

`:.\ text(Equation of tangents:)`

`y = 14x – 50`

`y = -2x – 2`

Calculus, 2ADV C4 2011 HSC 4c

The gradient of a curve is given by  `dy/dx = 6x-2`.  The curve passes through the point `(-1, 4)`. 

What is the equation of the curve?   (2 marks)

Show Answers Only

 `y = 3x^2-2x-1`

Show Worked Solution

`dy/dx = 6x-2` 

`y` `= int 6x-2\ dx`
  `= 3x^2-2x + c`

 
`text{Since it passes through}\ (-1,4),`

`4` `= 3 (-1)^2-2(-1) + c`
`4` `= 3 + 2 + c`
`c` `= -1`

 
`:. y = 3x^2-2x-1`

Calculus, 2ADV C4 2013 HSC 16a

The derivative of a function `f(x)` is  `f^{′}(x) = 4x-3`.  The line  `y = 5x-7`  is tangent to the graph `f(x)`.

Find the function `f(x)`.   (3 marks) 

Show Answers Only

`f(x) = 2x^2-3x + 1`

Show Worked Solution

`text(Solution 1)`

`f^{′}(x)` `=4x-3`
`f(x)` `= int 4x-3\ dx`
  `= 2x^2-3x + c`

 
`text(Intersection when)`

`2x^2-3x + c = 5x-7`

`2x^2-8x + (7 + c) = 0`

  
`text(S)text(ince)\ \ y=5x-7\ \ text(is a tangent)\ => Delta =0`

`b^2-4ac` `=0`
`(-8)^2-[4xx2xx(7+c)]` `= 0`
`64-56-8c` `=8`
`8c` `=8`
`c` `=1`

 
`:.f(x) = 2x^2-3x + 1`

 
`text(Solution 2)`

`f^{′}(x)=4x-3`

`y=5x-7\ \ text{(Gradient = 5)}`

`=>4x-3` `=5`
`x` `=2`

 
`f(x) = 2x^2-3x + 1`

`f(x)\ text{passes through (2, 3)}`

`f(2)` `=2xx 2^2-3(2)+c`
`3` `=8-6+c`
`c` `=1`

 
`:.f(x) = 2x^2-3x + 1`