Exam-Style Questions.

1.	IB Studies

A function is given as \(f(x)=3x^2-6x+4+\frac3x,-2\le x \le 4, x\ne 0\).

(a) Find the derivative of the function.

(b) Find the coordinates of the local minimum point of \(f(x)\) in the given domain using your calculator.

Worked Solution

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2.	IB Studies

Consider the graph of the function \(f(x)=7-3x^2-x^3\)

(a) Label the local maximum as A on the graph.

(b) Label the local minimum as B on the graph.

(c) Write down the interval where \(f(x)>5\).

(d) Draw the tangent to the curve at \(x=-3\) on the graph.

(e) Write down the equation of the tangent at \(x=-3\).

Worked Solution

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3.	IB Studies

Consider the function \(f(x)=x^3-9x+2\).

(a) Sketch the graph of \(y=f(x)\) for \(-4\le x\le 4\) and \(-14\le y\le 14\) showing clearly the axes intercepts and local maximum and minimum points. Use a scale of 2 cm to represent 1 unit on the x-axis, and a scale of 1 cm to represent 2 units on the y-axis.

(b) Find the value of \(f(-1)\).

(c) Write down the coordinates of the y-intercept of the graph of \(f(x)\).

(d) Find \(f'(x)\).

(e) Find \(f'(-1)\)

(f) Explain what \(f'(-1)\) represents.

(g) Find the equation of the tangent to the graph of \(f(x)\) at the point where x is –1.

R and S are points on the curve such that the tangents to the curve at these points are horizontal. The x-coordinate of R is \(a\) , and the x-coordinate of S is \(b\) , \(b \gt a\).

(h) Write down the value of \(a\) ;

(i) Write down the value of \(b\).

(j) Describe the behaviour of \(f(x)\) for \(a \lt x \lt b\).

Worked Solution

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4.	IB Standard

The following diagram shows part of the graph of \(y=f (x)\)

The graph has a local maximum where \(x=- \frac23\), and a local minimum where \(x=4\).

sketch the graph of \(y=f'(x)\) for \(-4\le x \le 7\)

Write down the following in order from least to greatest: \(f(2),f'(4)\) and \(f''(4)\).

Worked Solution

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5.	IB Studies

Consider the function \(f(x)=\frac{20}{x^2}+kx\) where \(k\) is a constant and \(x\neq0\).

(a) Write down \(f'(x)\)

The graph of \(y = f(x)\) has a local minimum point at \(x=2\).

(b) Show that \(k=5\).

(c) Find \(f(1)\).

(d) Find \(f'(1)\).

(e) Find the equation of the normal to the graph of \(y=f(x)\) at the point where \(x=1\)
     Give your answer in the form \(ay+bx+c=0\) where \(a, b, c \in \mathbb{Z}\)

(f) Sketch the graph of \(y=f(x)\) , for \(-5\le x\le 10\) and \(-10\le y\le 50\).

(g) Write down the coordinates of the point where the graph of \(y=f(x)\) intersects the x-axis.

(h) State the values of \(x\) for which \(f(x)\) is decreasing.

Worked Solution

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