Basic Calculus
"Future Engineer! Limits and Derivatives - ang heart ng Calculus. Dito nasusubok ang pasensya at problem-solving skills mo. Master these concepts para sa college math at engineering!"
1 Understanding Limits
A limit describes what happens to a function as x approaches a particular value.
limx→a f(x) = L
"As x approaches a, f(x) approaches L"
One-Sided Limits
- Left-hand limit: limx→a⁻ f(x) (from the left)
- Right-hand limit: limx→a⁺ f(x) (from the right)
- Limit exists only if both sides equal
When Limits Don't Exist
- • Left ≠ Right (jump discontinuity)
- • f(x) → ±∞ (infinite discontinuity)
- • f(x) oscillates (sin(1/x) near 0)
2 Limit Theorems and Properties
Basic Limit Laws
- Sum: lim[f(x) + g(x)] = lim f(x) + lim g(x)
- Difference: lim[f(x) - g(x)] = lim f(x) - lim g(x)
- Product: lim[f(x) · g(x)] = lim f(x) · lim g(x)
- Quotient: lim[f(x)/g(x)] = lim f(x) / lim g(x)
- Constant: lim[c · f(x)] = c · lim f(x)
- Power: lim[f(x)]ⁿ = [lim f(x)]ⁿ
Special Limits to Memorize:
3 Techniques for Evaluating Limits
1. Direct Substitution
If f(a) exists and is defined, just plug in the value!
2. Factoring (0/0 Form)
Factor and cancel common terms
3. Rationalization
Multiply by conjugate when radicals are involved
Useful for √ in numerator or denominator
4. L'Hôpital's Rule (0/0 or ∞/∞)
limx→a f(x)/g(x) = limx→a f'(x)/g'(x)
Take derivatives of top and bottom separately
4 Continuity of Functions
Three Conditions for Continuity at x = a:
- f(a) exists (function is defined at a)
- limx→a f(x) exists (limit exists)
- limx→a f(x) = f(a) (limit equals function value)
Removable
Hole in graph. Can be "fixed" by defining f(a).
Jump
Left and right limits exist but are different.
Infinite
Vertical asymptote. Limit is ±∞.
5 Introduction to Derivatives
The derivative measures the instantaneous rate of change (slope at a point).
f'(x) = limh→0 [f(x+h) - f(x)] / h
Definition of the derivative
Notations for Derivative
- f'(x) - Lagrange notation
- dy/dx - Leibniz notation
- Df(x) - D notation
- ẏ - Newton notation (physics)
Interpretations
- Geometry: Slope of tangent line
- Physics: Velocity = ds/dt
- Economics: Marginal cost/revenue
- Rate: Instantaneous rate of change
6 Differentiation Rules - Must Memorize!
| Rule | Formula | Example |
|---|---|---|
| Constant | d/dx(c) = 0 | d/dx(5) = 0 |
| Power Rule | d/dx(xⁿ) = nxⁿ⁻¹ | d/dx(x³) = 3x² |
| Sum Rule | (f+g)' = f' + g' | d/dx(x²+x) = 2x+1 |
| Product Rule | (fg)' = f'g + fg' | d/dx(x·sin x) |
| Quotient Rule | (f/g)' = (f'g-fg')/g² | d/dx(sin x/x) |
| Chain Rule | d/dx[f(g(x))] = f'(g(x))·g'(x) | d/dx(sin 2x) = 2cos 2x |
7 Common Derivatives Table
Trigonometric Functions
- d/dx(sin x) = cos x
- d/dx(cos x) = -sin x
- d/dx(tan x) = sec²x
- d/dx(cot x) = -csc²x
- d/dx(sec x) = sec x tan x
- d/dx(csc x) = -csc x cot x
Exponential & Logarithmic
- d/dx(eˣ) = eˣ
- d/dx(aˣ) = aˣ ln a
- d/dx(ln x) = 1/x
- d/dx(logₐx) = 1/(x ln a)
Memory Tip for Trig Derivatives:
"co" functions have negative derivatives. (cos, cot, csc all have negative signs)
Practice Questions
1. Evaluate: limx→2 (x²-4)/(x-2)
Show Answer
= 4. Factor: (x+2)(x-2)/(x-2) = x+2. Substitute x=2: 2+2 = 4.
2. Find d/dx(3x⁴ - 2x² + 5x - 1)
Show Answer
= 12x³ - 4x + 5. Use power rule on each term: 4(3)x³ - 2(2)x + 5 - 0.
3. Find d/dx(sin 3x)
Show Answer
= 3cos 3x. Using chain rule: cos(3x) · d/dx(3x) = cos(3x) · 3 = 3cos 3x.
4. Is f(x) = |x| continuous at x = 0? Is it differentiable at x = 0?
Show Answer
Continuous: YES (no break). Differentiable: NO (sharp corner - left slope = -1, right slope = +1).
Exam Tips for Basic Calculus
- ✓ Power Rule is essential: d/dx(xⁿ) = nxⁿ⁻¹ - memorize and practice!
- ✓ Chain Rule: "Derivative of outside × derivative of inside"
- ✓ 0/0 forms: Factor, rationalize, or use L'Hôpital's Rule
- ✓ Continuity vs Differentiability: Differentiable → Continuous, but NOT vice versa
- ✓ Memorize trig derivatives: sin→cos, cos→-sin, tan→sec²
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