Exponentielle – exercices de maths 1ère

Exercice 1Algèbre

1. \(e^4 \times e^{-7}\)
2. \(\dfrac{e^{3x}}{e^{x + 2}}\)
3. \((e^{2x})^3 \times e^{-4x}\)
4. \(\dfrac{e^{2x} - e^{x}}{e^{x}}\)
5. \((e^x + 1)^2 - (e^x - 1)^2\)

Exercice 2Équations

1. \(e^{x} = e^{-2}\)
2. \(e^{3x + 2} = 1\)
3. \(e^{x^2} = e^{9}\)
4. \(e^{4x} = e^{x + 6}\)
5. \(e^{x} + 3 = 0\)

Exercice 3Inéquations

1. \(e^{x} \geq e^{5}\)
2. \(e^{-2x} < 1\)
3. \(e^{3x - 1} \leq e^{x + 5}\)
4. \(e^{x^2 - 3x} > e^{4}\)

Exercice 4Dérivées

1. \(5e^{x} + x^3\)
2. \(e^{-3x + 2}\)
3. \(x^2e^{x}\)
4. \(\dfrac{e^{x}}{x + 1}\)
5. \(e^{x^2 + x}\)

Exercice 5Variations

1. \((x + 2)e^{x}\)
2. \(e^{3x} - 3x\)
3. \(x^2e^{-x}\)

Exercice 6Signe

1. \((3x - 4)e^{x}\)
2. \(1 - e^{x}\)
3. \((x^2 - 9)e^{-2x}\)

Exercice 7Suites

\(u_n = e^{0{,}5n}\).

1. Géométrique ? Raison ?
2. Variations ?
3. \(v_n = 3e^{-0{,}4n}\) ?

Exercice 8Tangente

1. Tangente à \(e^x\) en \(0\) ? en \(-1\) ?
2. \(e^{x} \geq x + 1\) via \(g(x) = e^{x} - x - 1\) ?

Exercice 9Croissance

\(N(t) = 800e^{0{,}25t}\).

1. \(N(0)\), \(N(4)\) ?
2. Facteur horaire ?
3. \(N'(t)\) ?

Exercice 10Problème

\(f(x) = (x^2 - 5x + 3)e^{x}\).

1. \(f'(x)\) factorisé ?
2. Variations ?
3. Tangente en \(0\) ?
4. \(f(x) = 0\) ?