Limits and Derivatives

Calculus is the branch of mathematics that studies change, and this chapter introduces its two basic ideas: limits and derivatives. It begins with a freely falling body, s = 4.9t^2, whose instantaneous velocity is found by shrinking the time interval. You will then learn the meaning of the limit of a function as x approaches a value, left-hand and right-hand limits, and when a limit exists. You will also study the algebra of limits and limits of polynomial and rational functions. You will study important standard limits: the limit of (x^n - a^n)/(x - a) as x -> a is n a^(n-1), the limit of sin x / x as x -> 0 is 1, and the limit of (1 - cos x)/x as x -> 0 is 0. The chapter then defines the derivative of a function at a point and finds derivatives from first principles. It covers the algebra of derivatives, including the sum, difference, product (Leibnitz) and quotient rules, and the derivatives of polynomials and trigonometric functions such as sin x, cos x and tan x.

Detailed Worksheet: Limits and Derivatives Section A - Definitions (10 marks) 1. Define the derivative of a function f at a point from first principles. (2 marks) 2. Evaluate the limit of (x^2 - 4)/(x - 2) as x -> 2. (2 marks) 3. Evaluate the limit of sin 4x / sin 2x as x -> 0. (2 marks) 4. State the product rule for derivatives. (2 marks) 5. Find the derivative of x^5 - 3x^2 + 7 with respect to x. (2 marks) Section B - Calculations and Applications (15 marks) 6. Evaluate the limit of (x^15 - 1)/(x^10 - 1) as x -> 1. (3 marks) 7. Evaluate the limit of (sqrt(1 + x) - 1)/x as x -> 0. (3 marks) 8. Evaluate the limit of (cos 2x - 1)/(cos x - 1) as x -> 0. (3 marks) 9. Find the derivative of f(x) = 1/x from first principles. (3 marks) 10. A ball is dropped from a tower, and the distance fallen after t seconds is s = 4.9t^2 metres. Find its velocity at t = 2 s and t = 3 s using derivatives. (3 marks) Section C - Diagrams (10 marks) 11. Draw the graph of f(x) = x^2 and show the tangent at x = 1. Explain the derivative as the slope of the tangent. (4 marks) 12. Draw the graph of a function with different left-hand and right-hand limits at x = 0, such as f(x) = x/mod(x) for x not 0, and explain why the limit does not exist. (3 marks) 13. Draw the graphs of y = sin x and y = x near x = 0 and use them to explain why the limit of sin x / x is 1. (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. Find the derivative of sin x from first principles. (5 marks) 15. Find the derivatives of (x + 1)(x^2 - 2) using the product rule and (x^2 + 1)/(x - 1) using the quotient rule. (5 marks) 16. Let f(x) = 2x + 3 for x <= 0 and f(x) = 3(x + 1) for x > 0. Find the left-hand and right-hand limits at x = 0 and at x = 1, and state whether the limits exist. (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Show all steps. Write limits in the form 'limit of f(x) as x -> value'.