Solutions Manual Thomas Calculus Early Transcendentals 15th Edition By Joel
Hass, Christopher Heil, Maurice Weir, Przemyslaw Bogacki Chapters 1-19.
Part 1: Functions (Chapter 1)
Question 1
Which of the following represents the formula for the difference quotient used to
calculate the slope of a secant line?
A) (f(x) - f(a))/(x - a)
B) (f(a) - f(x))/(a - x)
C) (f(x + h) - f(x))/h
D) All of the above
Answer: D
Rationale: All options are algebraically equivalent forms of the difference
quotient. The standard definition of the derivative uses the limit as h→0 of
(f(x+h)-f(x))/h, but the slope of the secant line can also be expressed as (f(b)-
f(a))/(b-a) . Both represent the average rate of change.
Question 2
Given f(x)=x²-3x+2, find f(a+h).
,A) a² + h² - 3a - 3h + 2
B) a² + 2ah + h² - 3a - 3h + 2
C) a² + h² - 3a + 2
D) a² + 2ah + h² - 3a + 2
Answer: B
Rationale: f(a+h) = (a+h)² - 3(a+h) + 2 = (a² + 2ah + h²) - 3a - 3h + 2. The middle
term 2ah is crucial and often missed.
Question 3
Find the natural domain of f(x) = √(x-5)/(x²-9).
A) [5,∞) excluding x = 3 only
B) [5,∞)
C) [5,∞) excluding x = -3 and x = 3
D) (-∞,-3) ∪ (-3,3) ∪ (3,∞)
Answer: C
Rationale: The square root requires x-5 ≥ 0, so x ≥ 5. The denominator cannot be
zero, so x ≠ 3 and x ≠ -3. Since the domain starts at 5, x = -3 is automatically
excluded, but x = 3 is within [5,∞) and must be excluded.
,Question 4
If f(x) = x² and g(x) = 2x + 1, what is (f ∘ g)(x)?
A) 2x² + 1
B) 4x² + 4x + 1
C) 2x² + 4x + 1
D) (2x+1)²
Answer: B
Rationale: (f ∘ g)(x) = f(g(x)) = f(2x+1) = (2x+1)² = 4x² + 4x + 1. This tests function
composition, a foundational skill for chain rule in Chapter 3 .
Question 5
The function y = f(ax) where a > 1 represents:
A) A horizontal stretch by factor a
B) A vertical stretch by factor a
C) A horizontal compression by factor 1/a
D) A vertical compression by factor 1/a
, Answer: C
Rationale: For y = f(ax) with a > 1, the graph compresses horizontally toward the
y-axis by factor 1/a. The period or width decreases. This is a standard graph
scaling property covered in Section 1.2 .
Question 6
Which of the following functions is even?
A) f(x) = x³ + x
B) f(x) = x⁴ - 2x² + 1
C) f(x) = √(x) + 1
D) f(x) = e^x
Answer: B
Rationale: Even functions satisfy f(-x) = f(x). For x⁴ - 2x² + 1, f(-x) = (-x)⁴ - 2(-x)² + 1
= x⁴ - 2x² + 1 = f(x). Odd functions satisfy f(-x) = -f(x) (Option A).
Question 7
The period of the function y = sin(3x) is:
A) π/3
B) 2π/3
Hass, Christopher Heil, Maurice Weir, Przemyslaw Bogacki Chapters 1-19.
Part 1: Functions (Chapter 1)
Question 1
Which of the following represents the formula for the difference quotient used to
calculate the slope of a secant line?
A) (f(x) - f(a))/(x - a)
B) (f(a) - f(x))/(a - x)
C) (f(x + h) - f(x))/h
D) All of the above
Answer: D
Rationale: All options are algebraically equivalent forms of the difference
quotient. The standard definition of the derivative uses the limit as h→0 of
(f(x+h)-f(x))/h, but the slope of the secant line can also be expressed as (f(b)-
f(a))/(b-a) . Both represent the average rate of change.
Question 2
Given f(x)=x²-3x+2, find f(a+h).
,A) a² + h² - 3a - 3h + 2
B) a² + 2ah + h² - 3a - 3h + 2
C) a² + h² - 3a + 2
D) a² + 2ah + h² - 3a + 2
Answer: B
Rationale: f(a+h) = (a+h)² - 3(a+h) + 2 = (a² + 2ah + h²) - 3a - 3h + 2. The middle
term 2ah is crucial and often missed.
Question 3
Find the natural domain of f(x) = √(x-5)/(x²-9).
A) [5,∞) excluding x = 3 only
B) [5,∞)
C) [5,∞) excluding x = -3 and x = 3
D) (-∞,-3) ∪ (-3,3) ∪ (3,∞)
Answer: C
Rationale: The square root requires x-5 ≥ 0, so x ≥ 5. The denominator cannot be
zero, so x ≠ 3 and x ≠ -3. Since the domain starts at 5, x = -3 is automatically
excluded, but x = 3 is within [5,∞) and must be excluded.
,Question 4
If f(x) = x² and g(x) = 2x + 1, what is (f ∘ g)(x)?
A) 2x² + 1
B) 4x² + 4x + 1
C) 2x² + 4x + 1
D) (2x+1)²
Answer: B
Rationale: (f ∘ g)(x) = f(g(x)) = f(2x+1) = (2x+1)² = 4x² + 4x + 1. This tests function
composition, a foundational skill for chain rule in Chapter 3 .
Question 5
The function y = f(ax) where a > 1 represents:
A) A horizontal stretch by factor a
B) A vertical stretch by factor a
C) A horizontal compression by factor 1/a
D) A vertical compression by factor 1/a
, Answer: C
Rationale: For y = f(ax) with a > 1, the graph compresses horizontally toward the
y-axis by factor 1/a. The period or width decreases. This is a standard graph
scaling property covered in Section 1.2 .
Question 6
Which of the following functions is even?
A) f(x) = x³ + x
B) f(x) = x⁴ - 2x² + 1
C) f(x) = √(x) + 1
D) f(x) = e^x
Answer: B
Rationale: Even functions satisfy f(-x) = f(x). For x⁴ - 2x² + 1, f(-x) = (-x)⁴ - 2(-x)² + 1
= x⁴ - 2x² + 1 = f(x). Odd functions satisfy f(-x) = -f(x) (Option A).
Question 7
The period of the function y = sin(3x) is:
A) π/3
B) 2π/3
