Multiple Choice
Question 1
If f(x) = x sin(x), find f'(x).
A
cos(x)
B
sin(x) + x cos(x)
C
sin(x) - x cos(x)
D
x cos(x) - sin(x)
Solution: Using the product rule: f'(x) = (d/dx(x))sin(x) + x(d/dx(sin(x))) = (1)sin(x) + x(cos(x)) = sin(x) + x cos(x).
Question 2
What is the derivative of f(x) = tan(x)?
A
cot(x)
B
sec^2(x)
C
-csc^2(x)
D
sec(x)tan(x)
Solution: The derivative of tan(x) is a standard trigonometric derivative formula: d/dx(tan(x)) = sec^2(x).
Question 3
Differentiate f(x) = x^2 cos(x).
A
2x sin(x)
B
2x cos(x) - x^2 sin(x)
C
2x cos(x) + x^2 sin(x)
D
-2x sin(x)
Solution: Using product rule: f'(x) = (d/dx(x^2))cos(x) + x^2(d/dx(cos(x))) = 2x cos(x) + x^2(-sin(x)) = 2x cos(x) - x^2 sin(x).
Question 4
If f(x) = 2x^3 - 4x^2 + 5x, what is the second derivative f''(x)?
A
6x^2 - 8x + 5
B
12x - 8
C
12x
D
6x - 4
Solution: f'(x) = 6x^2 - 8x + 5. f''(x) = d/dx(6x^2 - 8x + 5) = 12x - 8.
Question 5
Find dy/dx for the curve x^2 + y^2 = 25.
A
x/y
B
-x/y
C
y/x
D
-y/x
Solution: Implicit differentiation: 2x + 2y(dy/dx) = 0. Solving for dy/dx gives 2y(dy/dx) = -2x, so dy/dx = -x/y.
Question 6
A function f(x) is defined such that lim_{x→3} f(x) = f(3). Which of the following MUST be true?
A
f is differentiable at x = 3.
B
f is continuous at x = 3.
C
The graph of f has a horizontal tangent at x = 3.
D
f'(3) exists.
Solution: The definition provided is the definition of continuity at a point: lim_{x→c} f(x) = f(c).
Question 7
What is the derivative of f(x) = sec(x)?
A
csc(x)cot(x)
B
sec(x)tan(x)
C
-sec(x)tan(x)
D
tan^2(x)
Solution: The derivative of sec(x) is a standard trigonometric identity: d/dx(sec(x)) = sec(x)tan(x).
Question 8
Given f(1.9) = 2.7, f(2) = 3, and f(2.1) = 3.3, estimate f'(2) using the symmetric difference quotient.
A
3
B
2
C
6
D
1.5
Solution: Symmetric difference quotient: f'(2) ~ (f(2.1) - f(1.9))/(2.1 - 1.9) = (3.3 - 2.7)/0.2 = 0.6/0.2 = 3.
Question 9
The average rate of change of the function f(x) = x² + 3x - 5 over the interval [1, 4] is:
A
5
B
6
C
7
D
8
Solution: The average rate of change is calculated as (f(4) - f(1)) / (4 - 1). f(4) = 16 + 12 - 5 = 23, f(1) = 1 + 3 - 5 = -1. Thus, (23 - (-1)) / 3 = 24 / 3 = 8.
Question 10
If the position of a particle is given by s(t) = t³ - 2t² + 4, the instantaneous velocity at t = 2 is:
A
4
B
8
C
12
D
16
Solution: v(t) = s'(t) = 3t^2 - 4t. v(2) = 3(4) - 8 = 12 - 8 = 4.
Question 11
Which of the following represents the derivative of f(x) = 1/x at x = 2 using the limit definition?
A
-1/4
B
-1/2
C
1/4
D
1/2
Solution: f'(x) = lim(h→0) [1/(x + h) - 1/x] / h = -1/x². At x = 2, f'(2) = -1/4.
Question 12
Given the table of values for f(x), estimate f'(2) if f(1.9) = 3.61, f(2) = 4, and f(2.1) = 4.41.
A
0.8
B
2
C
4
D
8
Solution: Symmetric difference quotient: f'(2) ~ (f(2.1) - f(1.9))/0.2 = (4.41 - 3.61)/0.2 = 0.8/0.2 = 4.
Question 13
The slope of the tangent line to the curve y = sin(x) at x = π/2 is:
A
-1
B
0
C
1
D
π/2
Solution: The derivative of sin(x) is cos(x). At x = π/2, cos(π/2) = 0, so the slope is 0.
Question 14
Which of the following functions is NOT differentiable at x = 0?
A
f(x) = |x|
B
f(x) = x²
C
f(x) = sin(x)
D
f(x) = e^x
Solution: f(x) = |x| has a corner at x = 0, making it non-differentiable there. The other functions are smooth and differentiable everywhere.
Question 15
The derivative of f(x) = x⁴ - 3x² + 5x - 7 is:
A
4x³ - 6x + 5
B
4x³ - 3x + 5
C
4x³ - 6x - 7
D
x³ - 6x + 5
Solution: Applying the power rule: d/dx(x⁴) = 4x³, d/dx(-3x²) = -6x, d/dx(5x) = 5, d/dx(-7) = 0. Combined: 4x³ - 6x + 5.
Question 16
The derivative of f(x) = x⁻³ + 2x⁻¹ is:
A
-3x⁻⁴ - 2x⁻²
B
3x⁻⁴ + 2x⁻²
C
-3x⁻⁴ + 2x⁻²
D
3x⁻⁴ - 2x⁻²
Solution: Applying the power rule: d/dx(x⁻³) = -3x⁻⁴, d/dx(2x⁻¹) = -2x⁻². Combined: -3x⁻⁴ - 2x⁻².
Question 17
The derivative of f(x) = 3sin(x) - 2cos(x) is:
A
3cos(x) + 2sin(x)
B
3cos(x) - 2sin(x)
C
-3cos(x) + 2sin(x)
D
-3cos(x) - 2sin(x)
Solution: d/dx(3sin(x)) = 3cos(x), d/dx(-2cos(x)) = 2sin(x). Combined: 3cos(x) + 2sin(x).
Question 18
The derivative of f(x) = tan(x) is:
A
sec²(x)
B
csc²(x)
C
sec(x)tan(x)
D
-csc(x)cot(x)
Solution: The derivative of tan(x) is sec²(x).
Question 19
If f(x) = x⁴ - 2x³ + 5, then f''(x) is:
A
12x² - 12x
B
12x² - 6x
C
12x - 12
D
12x - 6
Solution: First derivative: f'(x) = 4x³ - 6x². Second derivative: f''(x) = 12x² - 12x.
Question 20
A particle moves along the x-axis such that its position at time t is given by x(t) = 5t^2 - t^3. What is the instantaneous velocity of the particle at t = 2?
A
4
B
8
C
-4
D
0
Solution: v(t) = x'(t) = 10t - 3t^2. v(2) = 20 - 12 = 8.