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Date May 2015 Marks available 2 Reference code 15M.2.sl.TZ2.5
Level SL only Paper 2 Time zone TZ2
Command term Find and Use 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 5x5 and 20y20.

[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×(2)282     (M1)

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

 

6     (A1)(G2)

a.

f(x)=x+8x2     (A1)(A1)(A1)

 

Notes: Award (A1) for x, (A1) for 8, (A1) for x2 or 1x2 (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=6ORy=0x+6ORy6=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, 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, 6)ORx=4, 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, 6)), follow through from the first point of intersection (to the right of 2).

Award (G0)(G0) for (2, 6).

g.

Examiners report

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

Syllabus sections

Topic 6 - Mathematical models » 6.7 » Use of a GDC to solve equations involving combinations of the functions above.
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