1.07g Differentiation from first principles: for small positive integer powers of x

7 \includegraphics[max width=\textwidth, alt={}, center]{17ca6dd2-271b-4b06-8433-354493feaf06-10_401_561_260_790} The diagram shows a rectangle \(A B C D\) in which \(A B = 5\) units and \(B C = 3\) units. Point \(P\) lies on \(D C\) and \(A P\) is an arc of a circle with centre \(B\). Point \(Q\) lies on \(D C\) and \(A Q\) is an arc of a circle with centre \(D\).

1. Show that angle \(A B P = 0.6435\) radians, correct to 4 decimal places.
2. Calculate the areas of the sectors \(B A P\) and \(D A Q\).
3. Calculate the area of the shaded region.

4 The function f is such that \(\mathrm { f } ^ { \prime \prime } ( x ) = \mathrm { f } ( x )\).
Prove by mathematical induction that, for every positive integer \(n\), $$\frac { d ^ { 2 n - 1 } } { d x ^ { 2 n - 1 } } ( x f ( x ) ) = x f ^ { \prime } ( x ) + ( 2 n - 1 ) f ( x )$$

7. \begin{figure}[h] \end{figure} Figure 4 shows part of the curve with equation \(y = 2 x ^ { 2 } + 5\) The point \(P ( 2,13 )\) lies on the curve.

1. Find the gradient of the tangent to the curve at \(P\). The point \(Q\) with \(x\) coordinate \(2 + h\) also lies on the curve.
2. Find, in terms of \(h\), the gradient of the line \(P Q\). Give your answer in simplest form.
3. Explain briefly the relationship between the answer to (b) and the answer to (a).

1. (a) By writing \(\operatorname { cosec } x\) as \(\frac { 1 } { \sin x }\), show that

$$\frac { \mathrm { d } ( \operatorname { cosec } x ) } { \mathrm { d } x } = - \operatorname { cosec } x \cot x$$ Given that \(y = \mathrm { e } ^ { 3 x } \operatorname { cosec } 2 x , 0 < x < \frac { \pi } { 2 }\),
(b) find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\). The curve with equation \(y = \mathrm { e } ^ { 3 x } \operatorname { cosec } 2 x , 0 < x < \frac { \pi } { 2 }\), has a single turning point.
(c) Show that the \(x\)-coordinate of this turning point is at \(x = \frac { 1 } { 2 } \arctan k\) where the value
of the constant \(k\) should be found. of the constant \(k\) should be found.

12

1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{15872003-2e41-47e9-a5bd-34e533768f8a-5_652_764_269_733} \captionsetup{labelformat=empty} \caption{Fig. 12}
   \end{figure} Fig. 12 shows part of the curve \(y = x ^ { 4 }\) and the line \(y = 8 x\), which intersect at the origin and the point P .
   (A) Find the coordinates of P , and show that the area of triangle OPQ is 16 square units.
   (B) Find the area of the region bounded by the line and the curve.
2. You are given that \(\mathrm { f } ( x ) = x ^ { 4 }\).
   (A) Complete this identity for \(\mathrm { f } ( x + h )\). $$f ( x + h ) = ( x + h ) ^ { 4 } = x ^ { 4 } + 4 x ^ { 3 } h + \ldots$$ (B) Simplify \(\frac { \mathrm { f } ( x + h ) - \mathrm { f } ( x ) } { h }\).
   (C) Find \(\lim _ { h \rightarrow 0 } \frac { \mathrm { f } ( x + h ) - \mathrm { f } ( x ) } { h }\).
   (D) State what this limit represents.

12

1. Calculate the gradient of the chord joining the points on the curve \(y = x ^ { 2 } - 7\) for which \(x = 3\) and \(x = 3.1\).
2. Given that \(\mathrm { f } ( x ) = x ^ { 2 } - 7\), find and simplify \(\frac { \mathrm { f } ( 3 + h ) - \mathrm { f } ( 3 ) } { h }\).
3. Use your result in part (ii) to find the gradient of \(y = x ^ { 2 } - 7\) at the point where \(x = 3\), showing your reasoning.
4. Find the equation of the tangent to the curve \(y = x ^ { 2 } - 7\) at the point where \(x = 3\).
5. This tangent crosses the \(x\)-axis at the point P . The curve crosses the positive \(x\)-axis at the point Q . Find the distance PQ , giving your answer correct to 3 decimal places.

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

1. Find the equation of the tangent to the curve \(y = x ^ { 4 }\) at the point where \(x = 2\). Give your answer in the form \(y = m x + c\).
2. Calculate the gradient of the chord joining the points on the curve \(y = x ^ { 4 }\) where \(x = 2\) and \(x = 2.1\).
3. (A) Expand \(( 2 + h ) ^ { 4 }\).
   (B) Simplify \(\frac { ( 2 + h ) ^ { 4 } - 2 ^ { 4 } } { h }\).
   (C) Show how your result in part (iii) (B) can be used to find the gradient of \(y = x ^ { 4 }\) at the point where \(x = 2\).
4. Calculate the gradient of the chord joining the points on the curve \(y = x ^ { 2 } - 7\) for which \(x = 3\) and \(x = 3.1\).
5. Given that \(\mathrm { f } ( x ) = x ^ { 2 } - 7\), find and simplify \(\frac { \mathrm { f } ( 3 + h ) - \mathrm { f } ( 3 ) } { h }\).
6. Use your result in part (ii) to find the gradient of \(y = x ^ { 2 } - 7\) at the point where \(x = 3\), showing your reasoning.
7. Find the equation of the tangent to the curve \(y = x ^ { 2 } - 7\) at the point where \(x = 3\).
8. This tangent crosses the \(x\)-axis at the point P . The curve crosses the positive \(x\)-axis at the point Q . Find the distance PQ , giving your answer correct to 3 decimal places.
9. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8ff8b67d-1489-4cb1-bcd2-b32db674e29f-3_651_770_242_737} \captionsetup{labelformat=empty} \caption{Fig. 12}
   \end{figure} Fig. 12 shows part of the curve \(y = x ^ { 4 }\) and the line \(y = 8 x\), which intersect at the origin and the point P .
   (A) Find the coordinates of P , and show that the area of triangle OPQ is 16 square units.
   (B) Find the area of the region bounded by the line and the curve.
10. You are given that \(\mathrm { f } ( x ) = x ^ { 4 }\).
    (A) Complete this identity for \(\mathrm { f } ( x + h )\). $$\mathrm { f } ( x + h ) = ( x + h ) ^ { 4 } = x ^ { 4 } + 4 x ^ { 3 } h + \ldots$$ (B) Simplify \(\frac { \mathrm { f } ( x + h ) - \mathrm { f } ( x ) } { h }\).
    (C) Find \(\lim _ { h \rightarrow 0 } \frac { \mathrm { f } ( x + h ) - \mathrm { f } ( x ) } { h }\).
    (D) State what this limit represents.

1

1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{1a6d059d-8ab8-41e0-8bf3-54e248f820e4-1_650_759_252_762} \captionsetup{labelformat=empty} \caption{Fig. 12}
   \end{figure} Fig. 12 shows part of the curve \(y = x ^ { 4 }\) and the line \(y = 8 x\), which intersect at the origin and the point P .
   (A) Find the coordinates of P , and show that the area of triangle OPQ is 16 square units.
   (B) Find the area of the region bounded by the line and the curve.
2. You are given that \(\mathrm { f } ( x ) = x ^ { 4 }\).
   (A) Complete this identity for \(\mathrm { f } ( x + h )\). $$\mathrm { f } ( x + h ) = ( x + h ) ^ { 4 } = x ^ { 4 } + 4 x ^ { 3 } h + \ldots$$ (B) Simplify \(\frac { \mathrm { f } ( x + h ) - \mathrm { f } ( x ) } { h }\).
   (C) Find \(\lim _ { h \rightarrow 0 } \frac { \mathrm { f } ( x + h ) - \mathrm { f } ( x ) } { h }\).
   (D) State what this limit represents.

6 \includegraphics[max width=\textwidth, alt={}, center]{debf6581-25ff-4692-bdfb-154675a3cdb0-3_608_1134_258_504} The diagram shows the curve \(y = \mathrm { f } ( x )\), defined by $$f ( x ) = \begin{cases} x ^ { x } & \text { for } 0 < x \leqslant 1 , \\ 1 & \text { for } x = 0 . \end{cases}$$

1. By first taking logarithms, show that the curve has a stationary point at \(x = \mathrm { e } ^ { - 1 }\). The area under the curve from \(x = 0.5\) to \(x = 1\) is denoted by \(A\).
2. By considering the set of three rectangles shown in the diagram, show that a lower bound for \(A\) is 0.388 .
3. By considering another set of three rectangles, find an upper bound for \(A\), giving 3 decimal places in your answer. The area under the curve from \(x = 0\) to \(x = 0.5\) is denoted by \(B\).
4. Draw a diagram to show rectangles which could be used to find lower and upper bounds for \(B\), using not more than three rectangles for each bound. (You are not required to find the bounds.)

6 Let \(\mathrm { f } ( x ) = 2 x ^ { 3 } + 3 x\). Use differentiation from first principles to show that \(\mathrm { f } ^ { \prime } ( x ) = 6 x ^ { 2 } + 3\).

3

1. Given that \(\mathrm { f } ( x ) = x ^ { 2 } + 2 x\), use differentiation from first principles to show that \(\mathrm { f } ^ { \prime } ( x ) = 2 x + 2\).
2. The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 x + 2\) and the curve passes through the point \(( - 1,5 )\). Find the equation of the curve.

3. Given that $$y = \frac { 1 } { 3 } x ^ { 3 }$$ use differentiation from first principle to show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = x ^ { 2 }$$

1. The curve \(C\) has equation

$$y = 2 x ^ { 3 } + 5$$ The curve \(C\) passes through the point \(P ( 1,7 )\).
Use differentiation from first principles to find the value of the gradient of the tangent to \(C\) at \(P\).

1. Given that

$$y = 2 x ^ { 2 }$$ use differentiation from first principles to show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 4 x$$

4 The quadratic polynomial \(2 x ^ { 2 } - 3\) is denoted by \(\mathrm { f } ( x )\).
Use differentiation from first principles to determine the value of \(\mathrm { f } ^ { \prime } ( 2 )\).

8

1. The quadratic polynomial \(a x ^ { 2 } + b x\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { f } ( x )\).
   Use differentiation from first principles to determine, in terms of \(a , b\) and \(x\), an expression for \(\mathrm { f } ^ { \prime } ( x )\).
2. \includegraphics[max width=\textwidth, alt={}, center]{a1f4ccbd-f5ed-437a-ae76-c4925ce86e25-07_565_1043_516_317} $$y = a x ^ { 2 } + b x$$ The diagram shows the quadratic curve \(y = a x ^ { 2 } + b x\), where \(a\) and \(b\) are constants. The shaded region is enclosed by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 4\). The tangent to the curve at \(x = 4\) intersects the \(x\)-axis at the point with coordinates \(( k , 0 )\).
   Given that the area of the shaded region is 9 units \({ } ^ { 2 }\), and the gradient of this tangent is \(- \frac { 3 } { 4 }\), determine the value of \(k\).

12 Below is a faulty argument that appears to show that the gradient of the curve \(y = x ^ { 2 }\) at the point \(( 3,9 )\) is 1 . Consider the chord joining \(( 3,9 )\) to the point \(\left( 3 + h , ( 3 + h ) ^ { 2 } \right)\) The gradient is \(\frac { ( 3 + h ) ^ { 2 } - 9 } { h } = \frac { 6 h + h ^ { 2 } } { h }\) When \(h = 0\) the gradient is \(\frac { 0 } { 0 }\) so the gradient of the curve is 1

1. Identify a fault in the argument.
2. Write a valid first principles argument leading to the correct value for the gradient at (3, 9).
3. Find the equation of the normal to the curve at the point ( 3,9 ).

6 Given that \(\mathrm { f } ( x ) = 2 x ^ { 2 } + 3\), show from first principles that \(\mathrm { f } ^ { \prime } ( x ) = 4 x\).

12 A function is defined by \(\mathrm { f } ( x ) = x ^ { 3 } - x\).

1. By considering \(\frac { f ( x + h ) - f ( x ) } { h }\), show from first principles that \(f ^ { \prime } ( x ) = 3 x ^ { 2 } - 1\).
2. Sketch the gradient function \(\mathrm { f } ^ { \prime } ( x )\).
3. Show that the curve \(y = f ( x )\) has a single point of inflection which is not a stationary point.

4 A point P on a piece of machinery is moving in a vertical straight line. The displacement of P above ground level at time \(t\) seconds is \(y\) metres. The displacement-time graph for the motion during the time interval \(0 \leqslant t \leqslant 4\) is shown in Fig. 7 . \begin{figure}[h] \end{figure}

1. Using the graph, determine for the time interval \(0 \leqslant t \leqslant 4\) (A) the greatest displacement of P above its position when \(t = 0\),
   (B) the greatest distance of P from its position when \(t = 0\),
   (C) the time interval in which P is moving downwards,
   (D) the times when P is instantaneously at rest. The displacement of P in the time interval \(0 \leqslant t \leqslant 3\) is given by \(y = - 4 t ^ { 2 } + 8 t + 12\).
2. Use calculus to find expressions in terms of \(t\) for the velocity and for the acceleration of P in the interval \(0 \leqslant t \leqslant 3\).
3. At what times does P have a speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the interval \(0 \leqslant t \leqslant 3\) ? In the time interval \(3 \leqslant t \leqslant 4 , \mathrm { P }\) has a constant acceleration of \(32 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). There is no sudden change in velocity when \(t = 3\).
4. Find an expression in terms of \(t\) for the displacement of P in the interval \(3 \leqslant t \leqslant 4\).

6

1. Expand \(( 5 + h ) ^ { 3 }\).
2. A curve has equation \(y = x ^ { 3 } - x ^ { 2 }\).
   1. Find the gradient of the line passing through the point \(( 5,100 )\) and the point on the curve for which \(x = 5 + h\). Give your answer in the form $$p + q h + r h ^ { 2 }$$ where \(p , q\) and \(r\) are integers.
   2. Show how the answer to part (b)(i) can be used to find the gradient of the curve at the point \(( 5,100 )\). State the value of this gradient.

2 A curve has equation \(y = x ^ { 4 } + x\).

1. Find the gradient of the line passing through the point \(( - 2,14 )\) and the point on the curve for which \(x = - 2 + h\). Give your answer in the form $$p + q h + r h ^ { 2 } + h ^ { 3 }$$ where \(p , q\) and \(r\) are integers.
2. Show how the answer to part (a) can be used to find the gradient of the curve at the point ( \(- 2,14\) ). State the value of this gradient.

5

1. Given that \(\mathrm { f } ( x ) = x ^ { 2 } - 4 x\), use differentiation from first principles to show that \(\mathrm { f } ^ { \prime } ( x ) = 2 x - 4\).
2. Find the equation of the curve through \(( 2,7 )\) for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 x - 4\).