Multiple Choice Questions
Calculus: Differentiation and Applications
Topic: Differentiation and Applications
Grade: 10
Question 1:
What is the derivative of f(x) = 3x^2 + 2x – 1?
a) 6x + 2
b) 3x + 2
c) 6x – 2
d) 3x – 2
Answer: a) 6x + 2
Explanation: To find the derivative of a polynomial function, we differentiate each term separately. The derivative of 3x^2 is 6x, the derivative of 2x is 2, and the derivative of -1 is 0. Therefore, the derivative of f(x) = 3x^2 + 2x – 1 is 6x + 2. For example, if we take x = 2, the slope of the tangent line to the graph of f(x) at x = 2 is 14.
Question 2:
Find the derivative of g(x) = (2x + 1)^3.
a) 12x^2 + 6x + 3
b) 6x^2 + 3x + 1
c) 6x^2 + 6x + 3
d) 12x^2 + 3x + 1
Answer: a) 12x^2 + 6x + 3
Explanation: To differentiate a function raised to a power, we use the chain rule. The derivative of (2x + 1)^3 is 3(2x + 1)^2 multiplied by the derivative of 2x + 1, which is 2. Simplifying this expression gives us 12x^2 + 6x + 3. For example, if we take x = 1, the slope of the tangent line to the graph of g(x) at x = 1 is 21.
Question 3:
What is the derivative of h(x) = sin(2x)?
a) 2cos(2x)
b) cos(2x)
c) -2cos(2x)
d) -sin(2x)
Answer: a) 2cos(2x)
Explanation: The derivative of sin(2x) can be found using the chain rule. The derivative of sin(u) is cos(u), and the derivative of 2x is 2. Therefore, the derivative of h(x) = sin(2x) is 2cos(2x). For example, if we take x = π/4, the slope of the tangent line to the graph of h(x) at x = π/4 is 2.
Question 4:
Find the derivative of f(x) = ln(3x^2 + 4x + 1).
a) (6x + 4)/(3x^2 + 4x + 1)
b) (3x + 2)/(3x^2 + 4x + 1)
c) (6x + 2)/(3x^2 + 4x + 1)
d) (3x + 4)/(3x^2 + 4x + 1)
Answer: b) (3x + 2)/(3x^2 + 4x + 1)
Explanation: To find the derivative of ln(u), where u is a function of x, we use the chain rule. The derivative of ln(u) is (1/u) multiplied by the derivative of u. In this case, u = 3x^2 + 4x + 1, so the derivative of f(x) = ln(3x^2 + 4x + 1) is (1/(3x^2 + 4x + 1)) multiplied by the derivative of 3x^2 + 4x + 1, which is (6x + 4). Simplifying this expression gives us (3x + 2)/(3x^2 + 4x + 1). For example, if we take x = 0, the slope of the tangent line to the graph of f(x) at x = 0 is 2.
Question 5:
What is the derivative of g(x) = e^(2x)?
a) 2e^(2x)
b) e^(2x)
c) 2e^(2x) + 1
d) e^(2x) + 1
Answer: a) 2e^(2x)
Explanation: The derivative of e^(2x) can be found using the chain rule. The derivative of e^u is e^u multiplied by the derivative of u. In this case, u = 2x, so the derivative of g(x) = e^(2x) is e^(2x) multiplied by the derivative of 2x, which is 2. Simplifying this expression gives us 2e^(2x). For example, if we take x = -1, the slope of the tangent line to the graph of g(x) at x = -1 is 2e^(-2).
(Note: More questions and explanations can be provided upon request)