Exam-Style Questions.

1.	IGCSE Extended

The table shows some values (rounded to one decimal place) for the function \(y=\frac{2}{x^2}-x, x\neq 0\).

\(x\)	-3	-2	-1	-0.5	0.5	1	2	3	4
\(y\)	3.2	2.5		8.5	7.5	1.0		-2.8

(a) Complete the table of values.

(b) Draw the graph of \(y=\frac{2}{x^2}-x\) for \(-3\le x \le -0.5\) and \(0.5\le x\le 4\).

(c) Use your graph to solve the equation \(\frac{2}{x^2}-x-3=0\)

(d) Use your graph to solve the equation \(\frac{2}{x^2}-x=1-2x\)

(e) By drawing a suitable tangent, find an estimate of the gradient of the curve at the point where x = 1.

(f) Using algebra, show that you can use the graph at \(y=0\) to find \(\sqrt[3]2\)

Worked Solution

---

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

---

3.	IB Studies

Consider the function \(f(x)=6 - ax+\frac 3{x^2},x\neq 0\)

(a) Write down the equation of the vertical asymptote of the graph of \(y=f(x)\).

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

The line T is the tangent to the graph of \(y=f(x)\) at the point where \(x=1\) and it has a gradient of -8.

(c) Show that \(a=2\).

(d) Find the equation of T.

(e) Using your calculator find the coordinates of the point where the graph of \(y=f(x)\) intersects the x-axis.

(f) The line T also intersects \(f(x)\) when \(-2\le x\le 0\). Find the coordinates of this intersection.

Worked Solution

---

4.	IB Analysis and Approaches

Consider the function \(f\) defined by \(f(x)= \ln{(x^2 - 9)}\) for \(x > 3\).

The following diagram shows part of the graph of \(f\) which crosses the x-axis at point \(A\) with coordinates \((a,0)\).

The line \(L\) is the tangent to the graph of \(f\) at the point \(B\) with coordinates \((b,c)\). The gradient of \(L\) is \( \frac14\)

(a) Find the exact value of \(a\).

(b) Find the value of \(b\).

Worked Solution

---

5.	IB Analysis and Approaches

The function \(f\) is defined for all \(x \in \mathbb{R}\). The line with equation \(y=5x+3\) is the tangent to the graph of \(f\) at \(x = 2\)

(a) Write down the value of \(f'(2)\).

(b) Find \(f(2)\).

The function \(g\) is defined for all \(x \in \mathbb{R}\) where \(g(x) = x^3 - 3x\) and \(h(x) = f(g(x)) \).

(c) Find \(h(2)\).

(d) Hence find the equation of the tangent to the graph of \(h\) at \(x = 2\).

Worked Solution

---

6.	IB Analysis and Approaches

Consider the cubic function \(f(x)=\frac{1}{6}x^3-2x^2+6x-2\)

(a) Find \(f'(x)\)

The graph of \(f\) has horizontal tangents at the points where \(x = a\) and \(x = b\) where \( a < b \).

(b) Find the value of \(a\) and the value of \(b\)

(c) Sketch the graph of \(y = f'(x)\).

(d) Hence explain why the graph of \(f\) has a local maximum point at \(x = a\).

(e) Find \(f''(b) \).

(f) Hence, use your answer to part (e) to show that the graph of \(f\) has a local minimum point at \(x = b\).

(g) Find the coordinates of the point where the normal to the graph of \(f\) at \(x = a\) and the tangent to the graph of \(f\) at \(x = b\) intersect.

Worked Solution

---

7.	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

---

8.	IB Standard

Let \(f(x)=\frac{g(x)}{h(x)}\), where \(g(3)=36\), \(h(3)=12\), \(g'(3)=10\) and \(h'(3)=4\). Find the equation of the normal to the graph of \(f\) at \(x=3\).

Worked Solution

---

9.	IB Analysis and Approaches

Consider the functions

$$f(x) = m-x^2 - 2xn - n^2$$ $$g(x) = \frac{e^x}{e^2} + \frac{m}{2}$$

where \(m, n \in \mathbb R\)

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

(b) Find \( g'(x)\).

The graphs of \(f\) and \(g\) have a common, non-vertical tangent at \(x=3\).

(c) Find expressions for \(m\) and \(n\) in terms of \(e\).

Worked Solution

---

10.	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

---