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:
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:
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:
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:
Exo 4: Using Substitution
Problem:
Find the integral of
f(x) = 3x^2 \cdot e^{x^3}
Solution:
Use substitution.
Let
u = x^3 \Rightarrow du = 3x^2 \, dx
The integral becomes:
\int 3x^2 e^{x^3} \, dx = \int e^u \, du = e^u + C
Substitute u = x^3
(No Ratings Yet)
Loading...