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)=0.5x2−8x, 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⩽ and - 20 \leqslant y \leqslant 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 \times {( - 2)^2} - \frac{8}{{ - 2}} (M1)
Note: Award (M1) for substitution of x = - 2 into the formula of the function.
6 (A1)(G2)
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.
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 = 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.
(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.
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,{\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).