smc-634-81-Tangents and transformations

Let `f: R to R, \ f(x)=x^{3}-x`.

Let `g_{a}: R to R` be the function representing the tangent to the graph of `f` at `x=a`, where `a in R`.

Let `(b, 0)` be the `x`-intercept of the graph of `g_{a}`.

Show that `b= {2a^{3}}/{3 a^{2}-1}`. (3 marks)
State the values of `a` for which `b` does not exist. (1 mark)
State the nature of the graph of `g_a` when `b` does not exist. (1 mark)
i. State all values of `a` for which `b=1.1`. Give your answer correct to four decimal places. (1 mark)
ii. The graph of `f` has an `x`-intercept at (1, 0).
State the values of `a` for which `1 <= b <= 1.1`.
Give your answers correct to three decimal places. (1 mark)

The coordinate `(b, 0)` is the horizontal axis intercept of `g_a`.

Let `g_b` be the function representing the tangent to the graph of `f` at `x=b`, as shown in the graph below.

Find the values of `a` for which the graphs of `g_a` and `g_b`, where `b` exists, are parallel and where `b!=a`. (3 marks)

Let `p:R rarr R, \ p(x)=x^(3)+wx`, where `w in R`.

Show that `p(-x)=-p(x)` for all `w in R`. (1 mark)

A property of the graphs of `p` is that two distinct parallel tangents will always occur at `(t, p(t))` and `(-t,p(-t))` for all `t!=0`.

Find all values of `w` such that a tangent to the graph of `p` at `(t, p(t))`, for some `t > 0`, will have an `x`-intercept at `(-t, 0)`. (1 mark)
Let `T:R^(2)rarrR^(2),T([[x],[y]])=[[m,0],[0,n]][[x],[y]]+[[h],[k]]`, where `m,n in R text(\{0})` and `h,k in R`.

State any restrictions on the values of `m`, `n`, `h`, and `k`, given that the image of `p` under the transformation `T` always has the property that parallel tangents occur at `x = -t` and `x = t` for all `t!=0`. (1 mark)

Show Answers Only

`text(See Worked Solutions.)`
`a=+-sqrt3/3`
`text(Horizontal line.)`
i. `a=-0.5052, 0.8084, 1.3468`
ii. `a in (-0.505,-0.500]uu(0.808,1.347)`
`a=+- sqrt5/5`
`text(See Worked Solutions.)`
`w=-5t^2`
`h=0`

Show Worked Solution

a. `f^(‘)(a) = 3a^2-1`

`g_a(x)\ \ text(has gradient)\ \ 3a^2-1\ \ text(and passes through)\ \ (a, a^3-a)`

`g_a(x)-(a^3-a)`	`=(3a^2- 1)(x-a)`
`g_a(x)`	`=(3a^2-1)(x-a)+a^3-a`

`x”-intercept occurs at”\ (b,0):`

`0=(3a^2-1)(b-a) + a^3-a`

`(3a^2-1)(b-a)`	`=a-a^3`
`3a^2b-3a^3-b+a`	`=a-a^3`
`b(3a^2-1)`	`=a-a^3+3a^3-a`
`:.b`	`=(2a^3)/(3a^2-1)`

b. `b\ text{does not exist when:}`

♦ Mean mark part (b) 46%.

`(3a^2-1)=0`

`a=+-sqrt3/3`

♦♦ Mean mark part (c) 23%.

c. `text{If}\ \ a=+-sqrt3/3,\ \ g_a^(‘)(x) = 0`

`=>\ text{the graph is a horizontal line (does not cross the}\ xtext{-axis).}`

d.i. `”Solve “{2a^{3}}/{3 a^{2}-1}=1.1” for “a:`

`a=-0.5052″ or “a=0.8084” or “a=1.3468\ \ text{(to 4 d.p.)}`

d.ii. `” Solve “1 <= (2a^(3))/(3a^(2)-1) < 1.1” for “a:`

♦♦♦ Mean mark part (d)(ii) 13%.

`a in (-0.505,-0.500]uu(0.808,1.347)\ \ text{(to 3 d.p.)}`

e. `f^(‘)(b) = 3b^2-1`

`g_b(x)\ \ text(has gradient)\ \ 3b^2-1\ \ text(and passes through)\ \ (b, b^3-b)`

`g_b(x)-(b^3-b)`	`=(3b^2- 1)(x-b)`
`g_b(x)`	`=(3b^2-1)(x-b)+b^3-b`

`g_a(x)\ text{||}\ g_b(x)\ \ text{when}`

♦♦♦ Mean mark part (e) 13%.

`3a^2-1`	`=3b^2-1`
	`=3(2a^3)/(3a^2-1)-1`

`=> a=+-1, +- sqrt5/5, 0`

`text(Test each solution so that)\ \ b!=a :`

`text(When)\ \ a=+-1, 0 \ => \ b=a`

`:. a=+- sqrt5/5`

f.	`p(-x)`	`=(-x)^3-wx`
		`=-x^3-wx`
		`=-(x^3+wx)`
		`=-p(x)`

g. `p^(‘)(t) = 3t^2+w`

♦♦♦ Mean mark part (g) 3%.

`p(t)\ \ text(has gradient)\ \ 3t^2+w\ \ text(and passes through)\ \ (t, t^3+wt)`

`p(t)-(t^3+wt)`	`=(3t^2+w)(x-t)`
`p(t)`	`=(3t^2+w)(x-t) + t^3+wt`

`text{If}\ p(t)\ text{passes through}\ \ (-t, 0):`

`0=(3t^2+w)(-2t) + t^3+wt`

`=>w=-5t^2\ \ (t>0)`

h. `text{Property of parallel tangents is retained under transformation}`

♦♦♦ Mean mark part (h) 2%.

`text{if rotational symmetry remains (odd function).}`

`=>h=0`

`text(No further restrictions apply to)\ m, n\ \ text{or}\ \ k.`

`y – 8`	`= 2 (x – 3)`
`y`	`= 2x + 2`

Calculus, MET2 2020 VCAA 5

Calculus, MET2 2009 VCAA 9 MC