UPCAT Mathematics — Calculus — Limits, Derivatives & IntegralsFlash Cards

Answer

Step 1: Apply the limit of a linear function theorem: lim(x→a) (mx + b) = ma + b
Step 2: Substitute a = 3, m = 2, b = 5
Step 3: lim(x→3) (2x + 5) = 2(3) + 5 = 6 + 5 = 11
Answer: 11

Answer

Step 1: Apply the power rule: d/dx(xⁿ) = nxⁿ⁻¹
Step 2: For f(x) = x⁴, we have n = 4
Step 3: f'(x) = 4x⁴⁻¹ = 4x³
Answer: f'(x) = 4x³

Answer

Step 1: Apply the power rule for integration: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C
Step 2: For 3x², factor out the constant: ∫ 3x² dx = 3∫ x² dx
Step 3: ∫ x² dx = x²⁺¹/(2+1) + C = x³/3 + C
Step 4: 3∫ x² dx = 3(x³/3) + C = x³ + C
Answer: x³ + C

Answer

Step 1: Direct substitution gives 0/0 (indeterminate form)
Step 2: Factor the numerator: x² - 4 = (x + 2)(x - 2)
Step 3: Simplify: (x² - 4)/(x - 2) = [(x + 2)(x - 2)]/(x - 2) = x + 2 (for x ≠ 2)
Step 4: Now evaluate: lim(x→2) (x + 2) = 2 + 2 = 4
Answer: 4

Answer

Step 1: Apply the sum/difference rule: derivative of sum = sum of derivatives
Step 2: d/dx(5x³) = 5 · 3x² = 15x² (constant multiple and power rule)
Step 3: d/dx(-2x) = -2 · 1 = -2 (constant multiple rule)
Step 4: d/dx(7) = 0 (derivative of constant)
Step 5: h'(x) = 15x² - 2 + 0 = 15x² - 2
Answer: h'(x) = 15x² - 2

Answer

Use the quotient rule when finding the derivative of f(x)/g(x) where both f(x) and g(x) are functions.
Formula: If h(x) = f(x)/g(x), then h'(x) = [g(x)f'(x) - f(x)g'(x)]/[g(x)]²
Example: For h(x) = x²/(x+1)
f(x) = x², f'(x) = 2x
g(x) = x+1, g'(x) = 1
h'(x) = [(x+1)(2x) - x²(1)]/(x+1)² = (2x² + 2x - x²)/(x+1)² = (x² + 2x)/(x+1)²

Answer

Step 1: Find the antiderivative F(x): ∫ 2x dx = 2 · x²/2 + C = x² + C
Step 2: Apply FTC Part 2: ∫ₐᵇ f(x)dx = F(b) - F(a)
Step 3: ∫₁³ 2x dx = [x²]₁³ = F(3) - F(1)
Step 4: F(3) = 3² = 9, F(1) = 1² = 1
Step 5: F(3) - F(1) = 9 - 1 = 8
Answer: 8

Answer

Step 1: Product rule: If h(x) = f(x)g(x), then h'(x) = f(x)g'(x) + f'(x)g(x)
Step 2: Let f(x) = 2x + 1, g(x) = x² - 3
Step 3: Find derivatives: f'(x) = 2, g'(x) = 2x
Step 4: Apply product rule:
h'(x) = (2x + 1)(2x) + (2)(x² - 3)
Step 5: Simplify: h'(x) = 4x² + 2x + 2x² - 6 = 6x² + 2x - 6
Answer: h'(x) = 6x² + 2x - 6

Answer

This is a standard limit that equals 1.
Step 1: This is a fundamental trigonometric limit
Step 2: lim(x→0) (sin x)/x = 1
Note: This limit is proven using the squeeze theorem and geometric arguments. It's a standard result used to derive other trigonometric limits.
Answer: 1

Answer

Step 1: Integrate term by term using sum/difference rule
Step 2: ∫ 3x² dx = 3 · x³/3 + C₁ = x³ + C₁
Step 3: ∫ -4x dx = -4 · x²/2 + C₂ = -2x² + C₂
Step 4: ∫ 5 dx = 5x + C₃
Step 5: Combine: ∫ (3x² - 4x + 5) dx = x³ - 2x² + 5x + C
Answer: x³ - 2x² + 5x + C

Answer

Step 1: A constant function f(x) = c has the same value for all x
Step 2: The derivative measures the rate of change
Step 3: Since a constant doesn't change, its rate of change is zero
Step 4: Using the definition: f'(x) = lim(h→0) [f(x+h) - f(x)]/h = lim(h→0) [c - c]/h = lim(h→0) 0/h = 0
Example: If f(x) = 7, then f'(x) = 0 because 7 never changes
Answer: Because constants have zero rate of change

Answer

Step 1: For limits at infinity with rational functions, divide by highest power of x
Step 2: Divide numerator and denominator by x:
lim(x→∞) (2x + 3)/(x - 1) = lim(x→∞) (2 + 3/x)/(1 - 1/x)
Step 3: As x→∞, 3/x→0 and 1/x→0
Step 4: lim(x→∞) (2 + 3/x)/(1 - 1/x) = (2 + 0)/(1 - 0) = 2/1 = 2
Answer: 2

Answer

Step 1: Rewrite using negative exponents: y = x⁻³ + 2x⁻¹
Step 2: Apply power rule: d/dx(xⁿ) = nxⁿ⁻¹
Step 3: d/dx(x⁻³) = -3x⁻³⁻¹ = -3x⁻⁴
Step 4: d/dx(2x⁻¹) = 2(-1)x⁻¹⁻¹ = -2x⁻²
Step 5: y' = -3x⁻⁴ - 2x⁻² = -3/x⁴ - 2/x²
Answer: y' = -3x⁻⁴ - 2x⁻² or -3/x⁴ - 2/x²

Answer

Step 1: Find antiderivative: ∫ (x² + 1) dx = x³/3 + x + C
Step 2: Apply FTC: ∫₀² (x² + 1) dx = [x³/3 + x]₀²
Step 3: Evaluate at upper limit x = 2: 2³/3 + 2 = 8/3 + 2 = 8/3 + 6/3 = 14/3
Step 4: Evaluate at lower limit x = 0: 0³/3 + 0 = 0
Step 5: Subtract: 14/3 - 0 = 14/3
Answer: 14/3

Answer

They are inverse operations (Fundamental Theorem of Calculus):
Step 1: If F(x) is the antiderivative of f(x), then F'(x) = f(x)
Step 2: If f(x) is the derivative of F(x), then ∫ f(x) dx = F(x) + C
Example: f(x) = 2x, F(x) = x²

• Derivative: d/dx(x²) = 2x ✓
• Integral: ∫ 2x dx = x² + C ✓

They "undo" each other like addition and subtraction
Answer: Derivatives and integrals are inverse operations

Answer

Step 1: Rewrite using fractional exponent: f(x) = √x = x^(1/2)
Step 2: Apply power rule: d/dx(xⁿ) = nxⁿ⁻¹
Step 3: f'(x) = (1/2)x^(1/2-1) = (1/2)x^(-1/2)
Step 4: Rewrite in radical form: f'(x) = (1/2)x^(-1/2) = 1/(2√x)
Step 5: Simplify: f'(x) = 1/(2√x)
Answer: f'(x) = 1/(2√x) or (1/2)x^(-1/2)

Answer

Step 1: Direct substitution gives 0/0 (indeterminate)
Step 2: Multiply by conjugate: (√x - 2)/(x - 4) · (√x + 2)/(√x + 2)
Step 3: Numerator: (√x - 2)(√x + 2) = x - 4
Step 4: Simplify: (x - 4)/[(x - 4)(√x + 2)] = 1/(√x + 2) for x ≠ 4
Step 5: Evaluate: lim(x→4) 1/(√x + 2) = 1/(√4 + 2) = 1/(2 + 2) = 1/4
Answer: 1/4

Answer

Step 1: Rewrite using negative exponent: 1/x² = x⁻²
Step 2: Apply power rule: ∫ xⁿ dx = x^(n+1)/(n+1) + C where n ≠ -1
Step 3: ∫ x⁻² dx = x^(-2+1)/(-2+1) + C = x⁻¹/(-1) + C
Step 4: Simplify: -x⁻¹ + C = -1/x + C
Answer: -1/x + C

Answer

Step 1: Product rule: d/dx[f(x)g(x)] = f(x)g'(x) + f'(x)g(x)
Step 2: Let f(x) = x² + 1, g(x) = 2x - 3
Step 3: Find derivatives: f'(x) = 2x, g'(x) = 2
Step 4: Apply formula:
d/dx[(x² + 1)(2x - 3)] = (x² + 1)(2) + (2x)(2x - 3)
Step 5: Expand: = 2x² + 2 + 4x² - 6x = 6x² - 6x + 2
Answer: 6x² - 6x + 2

Answer

Step 1: This is a PRODUCT of functions: x and eˣ
Step 2: ∫ [f(x)·g(x)] dx ≠ ∫ f(x) dx · ∫ g(x) dx
Step 3: Wrong approach: ∫ x dx · ∫ eˣ dx = (x²/2)·(eˣ) ❌
Step 4: Correct approach: Use integration by parts
Step 5: ∫ u dv = uv - ∫ v du where u = x, dv = eˣ dx
Remember: Integration distributes over ADDITION, not MULTIPLICATION
Answer: Because integration doesn't distribute over products - need integration by parts