Date | May 2021 | Marks available | 1 | Reference code | 21M.1.SL.TZ1.5 |
Level | Standard Level | Paper | Paper 1 (without calculator) | Time zone | Time zone 1 |
Command term | Find | Question number | 5 | Adapted from | N/A |
Question
Consider the functions f(x)=-(x-h)2+2k and g(x)=ex-2+k where h, k∈ℝ.
The graphs of f and g have a common tangent at x=3.
Find f'(x).
Show that h=e+62.
Hence, show that k=e+e24.
Markscheme
f'(x)=-2(x-h) A1
[1 mark]
g'(x)=ex-2 OR g'(3)=e3-2 (may be seen anywhere) A1
Note: The derivative of g must be explicitly seen, either in terms of x or 3.
recognizing f'(3)=g'(3) (M1)
-2(3-h)=e3-2 (=e)
-6+2h=e OR 3-h=-e2 A1
Note: The final A1 is dependent on one of the previous marks being awarded.
h=e+62 AG
[3 marks]
f(3)=g(3) (M1)
-(3-h)2+2k=e3-2+k
correct equation in k
EITHER
-(3-e+62)2+2k=e3-2+k A1
k=e+(6-e-62)2 (=e+(-e2)2) A1
OR
k=e+(3-e+62)2 A1
k=e+9-3e-18+e2+12e+364 A1
THEN
k=e+e24 AG
[3 marks]