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Date May 2015 Marks available 3 Reference code 15M.2.sl.TZ2.5
Level SL only Paper 2 Time zone TZ2
Command term Find Question number 5 Adapted from N/A

Question

Consider the function f(x)=0.5x28x, x0.

Find f(2).

[2]
a.

Find f(x).

[3]
b.

Find the gradient of the graph of f at x=2.

[2]
c.

Let T be the tangent to the graph of f at x=2.

Write down the equation of T.

[2]
d.

Let T be the tangent to the graph of f at x=2.

Sketch the graph of f for 5 and - 20 \leqslant y \leqslant 20.

[4]
e.

Let T be the tangent to the graph of f at x =  - 2.

Draw T on your sketch.

[2]
f.

The tangent, T, intersects the graph of f at a second point, P.

Use your graphic display calculator to find the coordinates of P.

[2]
g.

Markscheme

0.5 \times {( - 2)^2} - \frac{8}{{ - 2}}     (M1)

Note: Award (M1) for substitution of x =  - 2 into the formula of the function.

 

6     (A1)(G2)

a.

f'(x) = x + 8{x^{ - 2}}     (A1)(A1)(A1)

 

Notes: Award (A1) for x, (A1) for 8, (A1) for {x^{ - 2}} or \frac{1}{{{x^2}}} (each term must have correct sign). Award at most (A1)(A1)(A0) if there are additional terms present or further incorrect simplifications are seen.

b.

f'( - 2) =  - 2 + 8{( - 2)^{ - 2}}     (M1)

Note: Award (M1) for x =  - 2 substituted into their f'(x) from part (b).

 

= 0     (A1)(ft)(G2)

Note: Follow through from their derivative function.

c.

y = 6\;\;\;OR\;\;\;y = 0x + 6\;\;\;OR\;\;\;y - 6 = 0(x + 2)     (A1)(ft)(A1)(ft)(G2)

 

Notes: Award (A1)(ft) for their gradient from part (c), (A1)(ft) for their answer from part (a). Answer must be an equation.

Award (A0)(A0) for x = 6.

d.

     (A1)(A1)(A1)(A1)

 

Notes: Award (A1) for labels and some indication of scales in the stated window. The point (-2,{\text{ }}6) correctly labelled, or an x-value and a y-value on their axes in approximately the correct position, are acceptable indication of scales.

Award (A1) for correct general shape (curve must be smooth and must not cross the y-axis).

Award (A1) for x-intercept in approximately the correct position.

Award (A1) for local minimum in the second quadrant.

e.

Tangent to graph drawn approximately at x =  - 2     (A1)(ft)(A1)(ft)

 

Notes: Award (A1)(ft) for straight line tangent to curve at approximately x =  - 2, with approximately correct gradient. Tangent must be straight for the (A1)(ft) to be awarded.

Award (A1)(ft) for (extended) line passing through approximately their y-intercept from (d). Follow through from their gradient in part (c) and their equation in part (d).

f.

(4,{\text{ }}6)\;\;\;OR\;\;\;x = 4,{\text{ }}y = 6     (G1)(ft)(G1)(ft)

Notes: Follow through from their tangent from part (d). If brackets are missing then award (G0)(G1)(ft).

If line intersects their graph at more than one point (apart from ( - 2,{\text{ }}6)), follow through from the first point of intersection (to the right of - 2).

Award (G0)(G0) for ( - 2,{\text{ }}6).

g.

Examiners report

[N/A]
a.
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b.
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c.
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d.
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e.
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f.
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g.

Syllabus sections

Topic 7 - Introduction to differential calculus » 7.2 » The principle that f\left( x \right) = a{x^n} \Rightarrow f'\left( x \right) = an{x^{n - 1}} .
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