Limits and Derivatives quiz

The limit of (x^3 - 8)/(x - 2) as x -> 2 is:

The limit of sin x / x as x -> 0 is:

The derivative of x^4 is:

The derivative of sin x is:

The derivative of a constant is:

The limit of (1 - cos x)/x as x -> 0 is:

The derivative of tan x is:

If f(x) = 3x^2, then f'(2) is:

The limit of (x^2 + 3x) as x -> 1 is:

The derivative of cos x is:

Evaluate the limit of (x + 3) as x -> 3. (2 marks)

Evaluate the limit of (x^2 - 9)/(x - 3) as x -> 3. (2 marks)

Evaluate the limit of tan x / x as x -> 0. (2 marks)

Evaluate the limit of sin 3x / 5x as x -> 0. (2 marks)

Find the derivative of x^2 from first principles. (2 marks)

Find the derivative of 99x at x = 100. (2 marks)

Find the derivative of x^3 - 27 at x = 2. (2 marks)

Find the derivative of x^(-3) (5 + 3x). (2 marks)

Find the derivative of sin x cos x. (2 marks)

Find the derivative of (x^5 - cos x)/sin x. (2 marks)

Evaluate the limit of (x^4 - 1)/(x - 1) as x -> 1. (3 marks)

Evaluate the limit of (ax + b)/(cx + 1) as x -> 0. (3 marks)

A particle moves so that s = t^3 - 6t^2 + 9t metres. Find its velocity at t = 1 s and t = 4 s. (3 marks)

Find the derivative of 2x^3 - 5x^2 + 4x - 7 at x = 1. (3 marks)

The area of a circle is A = pi r^2. Find the rate of change of area with respect to radius when r = 5 cm (leave answer in terms of pi). (3 marks)

Read the passage and answer the questions. A stone is dropped from a cliff. The distance it falls in t seconds is s = 4.9t^2 metres. A student computes the average velocity over intervals from t = 2 to t = 2 + h for smaller and smaller h. (i) Find the average velocity from t = 2 to t = 2.1. (ii) Find the limit of the average velocity as h -> 0. (iii) What is this limit called? (5 marks)

Read the passage and answer the questions. A company's cost of producing x units is C(x) = 0.005x^3 - 0.02x^2 + 30x + 5000 rupees. The marginal cost is the derivative of C(x). (i) Find the marginal cost function. (ii) Find the marginal cost when 3 units are produced. (iii) What does marginal cost tell us? (5 marks)

Read the passage and answer the questions. A function is defined as f(x) = x^2 - 1 for x < 1 and f(x) = 2x for x >= 1. (i) Find the left-hand limit at x = 1. (ii) Find the right-hand limit at x = 1. (iii) Does the limit exist at x = 1? (5 marks)

Read the passage and answer the questions. The volume of a cube of side x is V = x^3. (i) Find dV/dx. (ii) Find the rate of change of volume with respect to side when x = 4 cm. (iii) Interpret the result. (5 marks)

Read the passage and answer the questions. A teacher asked students to find the limit of tan 2x / (x - pi/2) as x -> pi/2. She suggested substituting x = pi/2 + h. (i) Write tan 2x in terms of h. (ii) Find the limit. (iii) Name the standard limit used. (5 marks)

Explain the intuitive idea of a limit and of the left-hand and right-hand limits with examples. When does a limit exist? (6 marks)

State the algebra of limits. Prove that the limit of (x^n - a^n)/(x - a) as x -> a is n a^(n-1) for positive integers n. (6 marks)

Prove that the limit of sin x / x as x -> 0 is 1. Hence evaluate the limits of sin ax / bx and tan x / x as x -> 0. (6 marks)

Define the derivative. Find the derivatives of cos x and tan x from first principles. (6 marks)

State and prove the product rule for derivatives. Find the derivative of (x^2 + 1) cos x and of (sec x - 1)/(sec x + 1). (6 marks)