1. Limit Exercises
Exo 1: Basic Limit Calculation
Problem:
\[
\lim_{x \to 0} \frac{\sin(x)}{x}
\]
Solution:
This is a standard limit which is part of the limit formulas:
\lim_{x \to 0} \frac{\sin(x)}{x} = 1
\]
Exo 2: Using L’Hopital’s Rule
Problem:
\[
\lim_{x \to 0} \frac{1 – \cos(x)}{x}
\]
Solution:
This is a \(\frac{0}{0}\) indeterminate form, so we can apply L’Hopital’s Rule. We differentiate the numerator and the denominator separately:
Numerator: \(\frac{d}{dx}[1 – \cos(x)] = \sin(x)\)
Denominator: \(\frac{d}{dx}[x] = 1\)
Now, we compute the new limit:
\Rightarrow \lim_{x \to 0} \frac{\sin(x)}{1} = 0
\]
Exo 3: Limit at Infinity
Problem:
\[\lim_{x \to \infty} \frac{3x^2 + 5x}{x^2 – 4x + 1}\]
Solution:
For limits at infinity, we divide both the numerator and denominator by \(x^2\), the highest degree term:
\[\frac{3x^2 + 5x}{x^2 – 4x + 1} = \frac{3 + \frac{5}{x}}{1 – \frac{4}{x} + \frac{1}{x^2}}\]
As \(x \to \infty\), the terms with \(x\) in the denominator go to zero. Therefore:
\Rightarrow \lim_{x \to \infty} \frac{3 + 0}{1 + 0 + 0} = 3
\]
2. Derivative Exercises
Exo 1: Using the Power Rule
Problem:
Find the derivative of:
\[f(x) = 3x^4 – 5x^2 + 7x – 2\]
Solution:
Apply the power rule
\[\frac{d}{dx}[x^n] = n x^{n-1}\]
\[f'(x) = 3 \cdot 4x^3 – 5 \cdot 2x + 7\]
Exo 2: Product Rule
Problem:
Find the derivative of
\[f(x) = (2x^3)(x^2 – 1)\]
Solution:
Use the product rule:
\[\frac{d}{dx}[u(x) \cdot v(x)] = u'(x) v(x) + u(x) v'(x)\]
Let
\[u(x) = 2x^3 \Rightarrow u'(x) = 6x^2\]
\[v(x) = x^2 – 1 \Rightarrow v'(x) = 2x\]
Now apply the product rule:
\[f'(x) = (6x^2)(x^2 – 1) + (2x^3)(2x)\]
Simplify:
\[f'(x) = 6x^4 – 6x^2 + 4x^4\]
Exo 3: Chain Rule
Problem:
Find the derivative of
\[f(x) = \sin(3x^2)\]
Solution:
Use the chain rule:
\[\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)\]
Let
\[
u = 3x^2 \Rightarrow f(u) = \sin(u)
\]
\[
\frac{d}{dx}[\sin(u)] = \cos(u)
\]
\[
\frac{d}{dx}[3x^2] = 6x
\]
Now, applying the chain rule:
\[f'(x) = \cos(3x^2) \cdot 6x\]
3. Integration Exercises
Exo 1: Basic Power Rule
Problem:
Find the integral of
\[f(x) = 5x^3 – 2x^2 + 4x – 1\]
Solution:
Apply the power rule for integration:
\[\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \text{ (for } n \neq -1 \text{)}\]
\[\int (5x^3 – 2x^2 + 4x – 1) \, dx = \frac{5x^4}{4} – \frac{2x^3}{3} + \frac{4x^2}{2} – x + C\]
Exo 2: Integration of \(\frac{1}{x}\)
Find the integral of:
\[f(x) = \frac{1}{x}\]
Solution:
The integral of \(\frac{1}{x}\) is:
Exo 3: Trigonometric Integral
Problem:
Find the integral of
\[f(x) = \sin(x)\]
Solution:
The integral of \(\sin(x)\) is:
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