🧮 Unit 2 – Part A (2-Mark Q&A)

1. Continuity of Piecewise Function f(x) at x=2

Step 1: Set the Left-Hand Limit (LHL) equal to the Right-Hand Limit (RHL) at x=2
lim_x→2^- (cx² + 2x) = lim_x→2⁺ (x³ - cx)

Step 2: Substitute x=2 into both expressions:
c(2)² + 2(2) = (2)³ - c(2)
4c + 4 = 8 - 2c

Step 3: Solve the linear equation for c:
6c = 4
c = 4/6 = 2/3

2. Domain, Range, and Graph of f(x) = x²

Step 1 (Domain): Identify that f(x) is a polynomial, defined for all real numbers.
Domain: (-∞, ∞)

Step 2 (Range): Identify that x² is always non-negative.
Range: [0, ∞)

Step 3 (Graph): Sketch a parabola opening upward with its vertex at (0, 0).

3. Domain of f(x) = (2x⁴ - 5) / (x² + x - 6)

Step 1: Set the denominator equal to zero to find restricted values:
x² + x - 6 = 0

Step 2: Factor the quadratic equation:
(x + 3)(x - 2) = 0

Step 3: Solve for x to find the values excluded from the domain:
x = -3 and x = 2

Step 4: State the domain as all real numbers except the excluded values:
Domain: (-∞, -3) U (-3, 2) U (2, ∞)

4. Evaluate lim_x→1 (x² - 4x) / (x² - 3x - 4)

Step 1: Use direct substitution of x=1 into the numerator and denominator:
Numerator: (1)² - 4(1) = 1 - 4 = -3
Denominator: (1)² - 3(1) - 4 = 1 - 3 - 4 = -6

Step 2: Calculate the limit:
lim_x→1 ... = (-3) / (-6) = 1/2

5. Evaluate lim_x→5 (2x² - 3x + 4)

Step 1: Use direct substitution of x=5 (since it's a polynomial):
lim_x→5 (2x² - 3x + 4) = 2(5)² - 3(5) + 4

Step 2: Calculate the value:
2(25) - 15 + 4 = 50 - 15 + 4 = 39

6. Domain of f(x) = √(3-x) - √(2+x)

Step 1: Set the radicand (expression inside the root) of the first term ≥ 0:
3 - x ≥ 0 ⇒ x ≤ 3

Step 2: Set the radicand of the second term ≥ 0:
2 + x ≥ 0 ⇒ x ≥ -2

Step 3: Find the intersection of the two inequalities:
-2 ≤ x ≤ 3
Domain: [-2, 3]

7. Find dy/dx for cos(xy) = 1 + sin y

Step 1: Differentiate both sides with respect to x using implicit differentiation.

Step 2: Apply the Product Rule to d/dx(xy)
-sin(xy) · d/dx(xy) = 0 + cos(y) · dy/dx
-sin(xy) · (y + x(dy/dx)) = cos(y)(dy/dx)

Step 3: Distribute and isolate terms with dy/dx on one side:
-y sin(xy) = x sin(xy)(dy/dx) + cos(y)(dy/dx)

Step 4: Factor out dy/dx and solve:
dy/dx = (-y sin(xy)) / (x sin(xy) + cos(y))

8. Horizontal Tangent for y = sec x / (1 + tan x)

Step 1: Find dy/dx using the Quotient Rule and simplify the numerator:
dy/dx = [(sec x tan x)(1 + tan x) - sec x(sec²x)] / (1 + tan x)²
dy/dx = [sec x (tan x - 1)] / (1 + tan x)²

Step 2: Set the numerator to zero (for a horizontal tangent):
sec x (tan x - 1) = 0

Step 3: Solve for x:
sec x = 0 (No solution)
tan x - 1 = 0 ⇒ tan x = 1

Step 4: State the general solution for tan x = 1:
x = π/4 + nπ, where n is an integer.

9. Show Continuity of f(x) at x=1

Step 1: Calculate the Left-Hand Limit (LHL) at x=1:
lim_x→1^- (1 - x²) = 1 - (1)² = 0

Step 2: Calculate the Right-Hand Limit (RHL) at x=1:
lim_x→1⁺ (log x) = log(1) = 0

Step 3: Calculate the function value f(1):
f(1) = 1 - (1)² = 0

Step 4: Conclude continuity since LHL = RHL = f(1)
0 = 0 = 0
The function is continuous.

10. Find Critical Values of f(x) = 5x³ - 6x

Step 1: Find the first derivative f'(x):
f'(x) = 15x² - 6

Step 2: Set the derivative equal to zero:
15x² - 6 = 0

Step 3: Solve for x:
15x² = 6
x² = 6/15 = 2/5
x = ±√(2/5)