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)=0.5x2−8x, x≠0.
Find f(−2).
Find f′(x).
Find the gradient of the graph of f at x=−2.
Let T be the tangent to the graph of f at x=−2.
Write down the equation of T.
Let T be the tangent to the graph of f at x=−2.
Sketch the graph of f for −5⩽x⩽5 and −20⩽y⩽20.
Let T be the tangent to the graph of f at x=−2.
Draw T on your sketch.
The tangent, T, intersects the graph of f at a second point, P.
Use your graphic display calculator to find the coordinates of P.
Markscheme
0.5×(−2)2−8−2 (M1)
Note: Award (M1) for substitution of x=−2 into the formula of the function.
6 (A1)(G2)
f′(x)=x+8x−2 (A1)(A1)(A1)
Notes: Award (A1) for x, (A1) for 8, (A1) for x−2 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.
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.
y=6ORy=0x+6ORy−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.
(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.
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).
(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).