The Schools of the Association of Islamic Charitable Projects
Class : LS
Name:
Date : 2 / 5 / 2023
Duration :60 mins
Math Test
Part A:
Let 𝑓 be the function defined over ℝ by 𝑓 (𝑥) = 2𝑒 𝑥 − 𝑥 − 2 and designate by (𝐶) its
representative curve in an orthonormal system (𝑂; 𝑖⃗; 𝑗⃗) .
1) Determine
𝑙𝑖𝑚
𝑙𝑖𝑚
𝑓 (𝑥) 𝑎𝑛𝑑
𝑓 (𝑥)
𝑥 → +∞
𝑥 → −∞
2) a. Show that the straight line (𝑑) of equation 𝑦 = −𝑥 − 2 is an asymptote to (𝐶) at −∞
b. Prove that (𝐶) is above (𝑑) .
3) Study the variations of 𝑓 and draw its table of variations.
4) a. Show that the equation 𝑓(𝑥) = 0 admits two roots 0 and 𝛼 and that −1.6 < 𝛼 < −1.5 .
b. Study according to the values of 𝑥, the sign of 𝑓.
5) Trace (𝐶) and (𝑑) .
Part B:
Let 𝑔 be a function defined over ℝ by 𝑔 (𝑥) = 𝑒 2𝑥 − (𝑥 + 1)𝑒 𝑥 .
1) Prove that 𝑔′ (𝑥) = 𝑒 𝑥 𝑓(𝑥).
2) Set up the table of variations of the function 𝑔. ( you do not need to calculate the limits).
Part C:
Let ℎ be a function defined by ℎ (𝑥) = ln[𝑓(𝑥)] + 1
1) Determine the domain of definition of ℎ.
2) Study the variation of the function ℎ and set up its table of variations.
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