Basic Calculus
Limits, Derivatives, and Integration Fundamentals
In This Lesson
Limits
Definition
The limit of f(x) as x approaches a is L, written as: lim(x→a) f(x) = L
This means f(x) gets arbitrarily close to L as x gets close to a (but not equal to a).
Limit Laws
- lim(x→a) c = c (constant)
- lim(x→a) x = a
- lim(x→a) [f(x) ± g(x)] = lim f(x) ± lim g(x)
- lim(x→a) [f(x) × g(x)] = lim f(x) × lim g(x)
- lim(x→a) [f(x) / g(x)] = lim f(x) / lim g(x), if lim g(x) ≠ 0
Special Limits
- lim(x→0) (sin x)/x = 1
- lim(x→0) (1 - cos x)/x = 0
- lim(x→∞) (1 + 1/x)^x = e
Indeterminate Forms
When direct substitution gives these forms, use algebraic techniques:
0/0, ∞/∞, 0×∞, ∞-∞, 0⁰, 1^∞, ∞⁰
Continuity
Definition of Continuity
A function f(x) is continuous at x = a if:
- f(a) is defined
- lim(x→a) f(x) exists
- lim(x→a) f(x) = f(a)
Types of Discontinuity
- Removable: Limit exists but ≠ f(a)
- Jump: Left and right limits differ
- Infinite: Limit is ±∞
Continuous Functions
- Polynomials (everywhere)
- Rational functions (where defined)
- Trigonometric functions
- Exponential & logarithmic
Derivatives
Definition
f'(x) = lim(h→0) [f(x+h) - f(x)] / h
The derivative represents the instantaneous rate of change (slope of tangent line).
Basic Derivative Rules
- d/dx (c) = 0 (constant rule)
- d/dx (xⁿ) = nxⁿ⁻¹ (power rule)
- d/dx (cf) = c × f' (constant multiple)
- d/dx (f ± g) = f' ± g' (sum/difference)
Product & Quotient Rules
- Product: (fg)' = f'g + fg'
- Quotient: (f/g)' = (f'g - fg') / g²
Chain Rule
d/dx [f(g(x))] = f'(g(x)) × g'(x)
Derivative of outer × derivative of inner
Common Derivatives
- d/dx (sin x) = cos x
- d/dx (cos x) = -sin x
- d/dx (tan x) = sec²x
- d/dx (eˣ) = eˣ
- d/dx (ln x) = 1/x
- d/dx (aˣ) = aˣ ln a
Applications of Derivatives
Finding Extrema
- Find critical points where f'(x) = 0 or undefined
- Use first derivative test or second derivative test
- f''(x) > 0 → local minimum; f''(x) < 0 → local maximum
Related Rates
Problems involving rates of change of related quantities. Use implicit differentiation with respect to time.
Concavity & Inflection Points
- f''(x) > 0 → concave up (cup shape)
- f''(x) < 0 → concave down (cap shape)
- Inflection point: where concavity changes
Basic Integration
Antiderivative
F(x) is an antiderivative of f(x) if F'(x) = f(x). The indefinite integral is:
∫f(x)dx = F(x) + C
Basic Integration Rules
- ∫xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ -1)
- ∫1/x dx = ln|x| + C
- ∫eˣ dx = eˣ + C
- ∫sin x dx = -cos x + C
- ∫cos x dx = sin x + C
Definite Integral
∫[a to b] f(x)dx = F(b) - F(a)
Represents the area under the curve from x=a to x=b