19N.2.SL.TZ0.S_3

pestleMathematicsAASLPaper 219N· sl-5-3-differentiating-polynomials-n-e-zsource ↗

Let $f\left( x \right) = x - 8$ , $g\left( x \right) = {x^4} - 3$ and $h\left( x \right) = f\left( {g\left( x \right)} \right)$ .

Find $h\left( x \right)$ .

[2]

Let ${\text{C}}$ be a point on the graph of $h$ . The tangent to the graph of $h$ at ${\text{C}}$ is parallel to the graph of $f$ .

Find the $x$ -coordinate of ${\text{C}}$ .

[5]

Markscheme / solution

attempt to form composite (in any order) (M1)

eg $f\left( {{x^4} - 3} \right)$ , ${\left( {x - 8} \right)^4} - 3$

$h\left( x \right) = {x^4} - 11$ A1 N2

[2 marks]

recognizing that the gradient of the tangent is the derivative (M1)

eg ${h'}$

correct derivative (seen anywhere) (A1)

$h'\left( x \right) = 4{x^3}$

correct value for gradient of $f$ (seen anywhere) (A1)

$f'\left( x \right) = 1$ , $m = 1$

setting their derivative equal to $1$ (M1)

$4{x^3} = 1$

$0.629960$

$x = \sqrt[3]{{\frac{1}{4}}}$ (exact), $0.630$ A1 N3

[5 marks]

Examiners’ report

[N/A]