Calculus

Question 1

Given the function p(x) = x⁵ - 2x + 3, find the smallest value a such that p’(a) equals the average rate of change over the period [-2, 3]

3 marks

Question 2

Find the area under the curve over the interval [0, 4] of the function f(x) = √x + x² - 1

2 marks

Question 3

The volume of a cone with a hemispherical base is 756mm², using the formulae for the two areas, express the height in terms of the diametre. At what value d does the shape express the largest surface area?

5 marks

Question 4

Write a pseudocode that approximates a function’s x-intercepts using Neuton’s Method

5 marks

Question 5

The normal of a function f(c) = ac² + 2 is tangential to the curve at a, what value(s) could a be?

3 marks

Question 6

For what value(s) of c is the integral from [c, 2c²+3] of x⁻¹ not defined?

1 mark

Answer 6

c = 0

Question 7

What is the limiting value of y = - e⁻ˣ ÷ e² + 5a? Hence, write the range

2 marks

Answer 7

as x → ∞, y → 5a
⇒ Range {y ∈ ℝ: y < 5a}

Question 8

From First Principles, establish the derivative of y = √x + 2x

4 marks

Question 9

Graph the line that is perpendicular to the nth-derivative of y = x⁵ - 3x³ + x -3 , such that dⁿy/dxⁿ is in the form y = mx +c, at the point in which the derivative crosses the x-axis

4 marks

Question 10

What is the maximum y-value for the (n-2)th derivative of xⁿ + 2, {n ∈ ℕ: n > 4}?

3 marks

Question 11

Given the function g(x) = 3ax² + 2(a-3√b)x+, of which has the properties:g(2) = 10 and g’(4) = - 3, find the values of a and b

3 marks