1.07i Differentiate x^n: for rational n and sums

1. Differentiate with respect to \(x\)

$$3 x ^ { 2 } - \sqrt { x } + \frac { 1 } { 2 x }$$

1. The curve with equation \(y = \sqrt { 8 x }\) passes through the point \(A\) with \(x\)-coordinate 2 .

Find an equation for the tangent to the curve at \(A\).

1. Given that

$$y = \frac { x ^ { 4 } - 3 } { 2 x ^ { 2 } } ,$$

1. find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\),
2. show that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \frac { x ^ { 4 } - 9 } { x ^ { 4 } }\).

1 Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(y = x ^ { 6 } + \sqrt { x }\).

1 Differentiate \(6 x ^ { \frac { 5 } { 2 } } + 4\).

5 A is the point \(( 2,1 )\) on the curve \(y = \frac { 4 } { x ^ { 2 } }\).
B is the point on the same curve with \(x\)-coordinate 2.1.

1. Calculate the gradient of the chord AB of the curve. Give your answer correct to 2 decimal places.
2. Give the \(x\)-coordinate of a point C on the curve for which the gradient of chord AC is a better approximation to the gradient of the curve at A .
3. Use calculus to find the gradient of the curve at A .

1 Differentiate \(10 x ^ { 4 } + 12\).

10 \begin{figure}[h] \end{figure} Fig. 10 shows a solid cuboid with square base of side \(x \mathrm {~cm}\) and height \(h \mathrm {~cm}\). Its volume is \(120 \mathrm {~cm} ^ { 3 }\).

1. Find \(h\) in terms of \(x\). Hence show that the surface area, \(A \mathrm {~cm} ^ { 2 }\), of the cuboid is given by \(A = 2 x ^ { 2 } + \frac { 480 } { x }\).
2. Find \(\frac { \mathrm { d } A } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } A } { \mathrm {~d} x ^ { 2 } }\).
3. Hence find the value of \(x\) which gives the minimum surface area. Find also the value of the surface area in this case.

1 Differentiate \(x + \sqrt { x ^ { 3 } }\).

4 A curve has equation \(y = x + \frac { 1 } { x }\).
Use calculus to show that the curve has a turning point at \(x = 1\).
Show also that this point is a minimum.

9 \begin{figure}[h] \end{figure} Fig. 9 shows a sketch of the graph of \(y = x ^ { 3 } - 10 x ^ { 2 } + 12 x + 72\).

1. Write down \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Find the equation of the tangent to the curve at the point on the curve where \(x = 2\).
3. Show that the curve crosses the \(x\)-axis at \(x = - 2\). Show also that the curve touches the \(x\)-axis at \(x = 6\).
4. Find the area of the finite region bounded by the curve and the \(x\)-axis, shown shaded in Fig. 9 . [4]

2 Given that \(y = 6 x ^ { \frac { 3 } { 2 } }\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
Show, without using a calculator, that when \(x = 36\) the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) is \(\frac { 3 } { 4 }\).

9 The equation of a cubic curve is \(y = 2 x ^ { 3 } - 9 x ^ { 2 } + 12 x - 2\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and show that the tangent to the curve when \(x = 3\) passes through the point \(( - 1 , - 41 )\).
2. Use calculus to find the coordinates of the turning points of the curve. You need not distinguish between the maximum and minimum.
3. Sketch the curve, given that the only real root of \(2 x ^ { 3 } - 9 x ^ { 2 } + 12 x - 2 = 0\) is \(x = 0.2\) correct to 1 decimal place.

6 Use calculus to find the \(x\)-coordinates of the turning points of the curve \(y = x ^ { 3 } - 6 x ^ { 2 } - 15 x\). Hence find the set of values of \(x\) for which \(x ^ { 3 } - 6 x ^ { 2 } - 15 x\) is an increasing function.

8 The gradient of a curve is \(3 \sqrt { x } - 5\). The curve passes through the point ( 4,6 ). Find the equation of the curve.

4 Find the equation of the tangent to the curve \(y = x ^ { 3 } + 2 x - 7\) at the point where it cuts the \(y\) axis.

9 The equation of a curve is given by \(y = ( x - 1 ) ^ { 2 } ( x + 2 )\).

1. Write \(( x - 1 ) ^ { 2 } ( x + 2 )\) in the form \(x ^ { 3 } + p x ^ { 2 } + q x + r\) where \(p , q\) and \(r\) are to be determined.
2. Show that the curve \(y = ( x - 1 ) ^ { 2 } ( x + 2 )\) has a maximum point when \(x = - 1\) and find the coordinates of the minimum point.
3. Sketch the curve \(y = ( x - 1 ) ^ { 2 } ( x + 2 )\).
4. For what values of \(k\) does \(( x - 1 ) ^ { 2 } ( x + 2 ) = k\) have exactly one root.

12 Fig. 12 shows a window. The base and sides are parts of a rectangle with dimensions \(2 x\) metres horizontally by \(y\) metres vertically. The top is a semicircle of radius \(x\) metres. The perimeter of the window is 10 metres. \begin{figure}[h] \end{figure}

1. Express \(y\) as a function of \(x\).
2. Find the total area, \(A \mathrm {~m} ^ { 2 }\), in terms of \(x\) and \(y\). Use your answer to part (i) to show that this simplifies to $$A = 10 x - 2 x ^ { 2 } - \frac { 1 } { 2 } \pi x ^ { 2 }$$
3. Prove that for the maximum value of \(A\), \(y = x\) exactly.
   \section*{MEI STRUCTURED MATHEMATICS } \section*{CONCEPTS FOR ADVANCED MATHEMATICS, C2} \section*{Practice Paper C2-B
   Insert sheet for question 11} 11 Speed-time graph with the first two points plotted. \includegraphics[max width=\textwidth, alt={}, center]{73d1c02b-1b7b-426d-a171-c762597cfed4-5_768_1772_1389_205}

4 You are given that \(y = x ^ { 3 } - 12 x\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Hence find the coordinates of the turning points of the curve.

9 The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } - 12 x + 9\). The curve passes through the point \(( 2 , - 2 )\).

1. Find the equation of the curve.
2. Show that the curve touches the \(x\)-axis at one point (A) and cuts it at another (B). State the coordinates of A and B.
3. The curve cuts the \(y\)-axis at C . Show that the tangent at C is perpendicular to the normal at B.

10 Fig. 10 shows the curve with equation \(y = x ^ { 2 } + \frac { 16 } { x }\). \begin{figure}[h] \end{figure}

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Hence calculate the coordinates of the stationary point on the curve.
3. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) and explain why this confirms that he stationary point is a minimum.
4. Using the trapezium rule with 4 intervals, estimate the area between the curve and the \(x\) axis between \(x = 2\) and \(x = 4\).
5. State, giving a reason, whether this estimate of the area under-estimates or over-estimates the true area beneath the curve.

9. The curve \(C\) has the equation \(y = \mathrm { f } ( x )\) where $$f ^ { \prime } ( x ) = 1 + \frac { 2 } { \sqrt { x } } , \quad x > 0$$ The straight line \(l\) has the equation \(y = 2 x - 1\) and is a tangent to \(C\) at the point \(P\).

1. State the gradient of \(C\) at \(P\).
2. Find the \(x\)-coordinate of \(P\).
3. Find an equation for \(C\).
4. Show that \(C\) crosses the \(x\)-axis at the point \(( 1,0 )\) and at no other point.

4 Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when

1. \(y = 2 x ^ { - 5 }\),
2. \(y = \sqrt [ 3 ] { x }\).

7 Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(y = \sqrt { x } + \frac { 3 } { x }\).

10 Given tha \(y = 6 x ^ { \frac { 3 } { 2 } }\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
Show, without using a calculator, that when \(x = 36\) the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) is \(\frac { 3 } { 4 }\).