IB Math AA HL Paper 3 Question Bank
1.) The family of curves y = f(x) = (ax3 + bx2 + cx + d)n, where a, b, c, d, and n are constants, is given.
a) Determine the following:
i) the domain and range of the function y = f(x) for any non-negative values of n.
The domain of the function y = f(x) is all real numbers because the polynomial is defined for all real values of x. The range of the function y = f(x) is all non-negative real numbers because the power of the polynomial is non-negative.
ii) Determine the x-intercepts of the function y = f(x) for any non-negative values of n.
To find the x-intercepts of the function, we set y = 0 and solve for x. Therefore, we need to find the roots of the polynomial ax3 + bx2 + cx + d = 0. We can use the factor theorem or the synthetic division to find the roots of the polynomial.
iii) Determine the y-intercept of the function y = f(x) for any non-negative values of n.
The y-intercept of the function y = f(x) is (0,dn) since the function is equal to dn when x = 0.
b) For n = 2, determine the inflection point of the function y = f(x).
For n = 2, the function becomes y = (ax3 + bx2 + cx + d)2. To find the inflection point, we need to find the second derivative of the function, which is:
f”(x) = 6a2x + 2b
Now we need to find the x values where the second derivative is equal to zero.
6a2x + 2b = 0
x = -b/(3a2)
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c) Next, Consider the curve, y2 = 2x3 + x
i) Find the derivative of y2 = 2x3 + x
y2 = 2x3 + x
y = √(2x3 + x)
Next, we can take the derivative of this expression using the chain rule. The chain rule states that if we have a function u(x) = f(g(x)), then the derivative of u(x) with respect to x is given by:
du/dx = df/dg * dg/dx
In this case, we have u(x) = sqrt(2x3 + x) and we have to find the derivative of this function with respect to x
du/dx = (1/2)(2x3 + x)(-1/2)(6x2 + 1) = (3x(2x2 + 1))/(2√(2x3 + x))
So the derivative of y = sqrt(2x3 + x) with respect to x is (3x(2x2 + 1))/(2√(2x3 + x))
ii) Hence, find the local minima or maxima of y2 = 2x3 + x
To find the local maxima or minima of y2 = 2x3 + x, we need to find the critical points of the function
y = √(2x3 + x).
A critical point of a function is a point where the derivative of the function is equal to zero or does not exist.
First, we need to find the derivative of y = √(2x3 + x) with respect to x:
y’ = (3x(2x2 + 1))/(2√(2x3 + x))
Next, we need to set y’ equal to zero and solve for x:
(3x(2x2 + 1))/(2√(2x3 + x)) = 0
3x(2x2 + 1) = 0
x = 0
Now, we need to check if x = 0 is a local maximum or minimum. To do this, we need to find the second derivative of y = √(2x3 + x) with respect to x:
y” = (6x(2x2 + 1) – 3(2x2 + 1)(2x))/ (2√(2x3 + x))2
We need to evaluate y” at x = 0, we get
y”(0) = -3/2, which is negative and that means that x = 0 is a local minimum.
So the local minima of y2 = 2x3 + x is (0, 0)
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