Calculus 1 Final Cheat Sheet
Introduction
Calculus 1 is the gateway to understanding change and motion. This cheat sheet is designed to condense key concepts, formulas, and strategies into a quick-reference guide for your final exam. Master these essentials to tackle problems with confidence.
1. Limits and Continuity
Key Formulas:
- Limit Laws:
[
\lim{x \to a} (f(x) \pm g(x)) = \lim{x \to a} f(x) \pm \lim{x \to a} g(x)
]
[
\lim{x \to a} c \cdot f(x) = c \cdot \lim{x \to a} f(x)
]
- Special Limits:
[
\lim{x \to 0} \frac{\sin x}{x} = 1, \quad \lim{x \to 0} \frac{e^x - 1}{x} = 1, \quad \lim{x \to 0} \frac{\ln(1 + x)}{x} = 1
]
- Continuity: A function ( f(x) ) is continuous at ( x = a ) if:
[
\lim_{x \to a} f(x) = f(a)
]
Pro Tip: Use algebraic manipulation or L’Hôpital’s Rule for indeterminate forms (( \frac{0}{0} ) or ( \frac{\infty}{\infty} )).
2. Derivatives
Basic Rules:
- Power Rule:
[
\frac{d}{dx} (x^n) = nx^{n-1}
]
- Product Rule:
[
\frac{d}{dx} [f(x)g(x)] = f’(x)g(x) + f(x)g’(x)
]
- Quotient Rule:
[
\frac{d}{dx} \left( \frac{f(x)}{g(x)} \right) = \frac{f’(x)g(x) - f(x)g’(x)}{[g(x)]^2}
]
- Chain Rule:
[
\frac{d}{dx} [f(g(x))] = f’(g(x)) \cdot g’(x)
]
Common Derivatives:
[
\frac{d}{dx} (e^x) = e^x, \quad \frac{d}{dx} (\ln x) = \frac{1}{x}, \quad \frac{d}{dx} (\sin x) = \cos x, \quad \frac{d}{dx} (\cos x) = -\sin x
]
Pro Tip: Always simplify before differentiating complex expressions.
3. Applications of Derivatives
Critical Points:
- Find ( f’(x) ) and set it equal to zero to locate critical points.
- Use the Second Derivative Test:
[
f”(x) > 0 \Rightarrow \text{Local Minimum}, \quad f”(x) < 0 \Rightarrow \text{Local Maximum}
]
Mean Value Theorem:
If ( f(x) ) is continuous on ([a, b]) and differentiable on ((a, b)), then:
[
\exists c \in (a, b) \text{ such that } f’© = \frac{f(b) - f(a)}{b - a}
]
Pro Tip: Sketch graphs to visualize behavior at critical points.
4. Integration
Basic Rules:
- Power Rule:
[
\int x^n \, dx = \frac{x^{n+1}}{n+1} + C, \quad n \neq -1
]
- Integration of ( \frac{1}{x} ):
[
\int \frac{1}{x} \, dx = \ln |x| + C
]
- Substitution Rule:
[
\int f(g(x)) \cdot g’(x) \, dx = \int f(u) \, du, \quad u = g(x)
]
Common Integrals:
[
\int e^x \, dx = e^x + C, \quad \int \sin x \, dx = -\cos x + C, \quad \int \cos x \, dx = \sin x + C
]
Pro Tip: Always check your integral by differentiating the result.
5. Applications of Integration
Area Between Curves:
[
\text{Area} = \int_{a}^{b} [f(x) - g(x)] \, dx
]
Volume of Solids (Disk Method):
[
V = \pi \int_{a}^{b} [f(x)]^2 \, dx
]
Pro Tip: Draw diagrams to understand the limits of integration.
6. Techniques of Integration
Integration by Parts:
[
\int u \, dv = uv - \int v \, du
]
Pro Tip: Choose ( u ) and ( dv ) using the acronym LIATE: Logs, Inverse Trig, Algebraic, Trig, Exponential.
Partial Fraction Decomposition:
Break down rational functions into simpler fractions:
[
\frac{P(x)}{Q(x)} = \sum \frac{A}{(x - r)^n}
]
Pro Tip: Clear denominators and equate coefficients to solve for constants.
7. Sequences and Series
Convergence Tests:
- Divergence Test: If ( \lim_{n \to \infty} an \neq 0 ), the series diverges.
- Ratio Test:
[
L = \lim{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \Rightarrow \begin{cases}
L < 1 & \text{Converges} \
L > 1 & \text{Diverges} \
L = 1 & \text{Inconclusive}
\end{cases}
]
Pro Tip: Always try the Divergence Test first; it’s the simplest.
FAQ Section
How do I determine if a limit exists at a point?
+Check if the left-hand limit, right-hand limit, and function value at the point are equal. If they match, the limit exists.
When should I use the Chain Rule?
+Use the Chain Rule when differentiating composite functions, such as f(g(x)) .
What’s the difference between \int f(x) \, dx and \frac{d}{dx} f(x) ?
+Integration ( \int f(x) \, dx ) is the reverse of differentiation ( \frac{d}{dx} f(x) ). Integration finds the function whose derivative is f(x) , while differentiation finds the rate of change of f(x) .
How do I choose between substitution and integration by parts?
+Use substitution when the integral contains a composite function and its derivative. Use integration by parts when the integral is a product of two functions.
Conclusion
Calculus 1 is a foundation for understanding change and accumulation. This cheat sheet covers the essentials, but practice is key. Review each concept, work through problems, and approach the final with confidence. Good luck!
