1.07h Differentiation from first principles: for sin(x) and cos(x)

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

6. The function f is defined by $$f ( x ) = \frac { 5 x - 3 } { x - 4 } \quad x > 4$$

1. Show, by using calculus, that f is a decreasing function.
2. Find \(\mathrm { f } ^ { - 1 }\)
3. 1. Show that \(\mathrm { ff } ( x ) = \frac { a x + b } { x + c }\) where \(a , b\) and \(c\) are constants to be found.
   2. Deduce the range of ff.

1. The function f is defined by

$$f ( x ) = \frac { 2 x ^ { 2 } - 32 } { 3 x ^ { 2 } + 7 x - 20 } + \frac { 8 } { 3 x - 5 } \quad x \in \mathbb { R } \quad x > 2$$

1. Show that \(\mathrm { f } ( x ) = \frac { 2 x } { 3 x - 5 }\)
2. Show, using calculus, that f is a decreasing function. You must make your reasoning clear. The function g is defined by $$g ( x ) = 3 + 2 \ln x \quad x \geqslant 1$$
3. Find \(\mathrm { g } ^ { - 1 }\)
4. Find the exact value of \(a\) for which $$\operatorname { gf } ( a ) = 5$$

1. (a) By writing \(\sec x\) as \(\frac { 1 } { \cos x }\), show that \(\frac { \mathrm { d } ( \sec x ) } { \mathrm { d } x } = \sec x \tan x\).

Given that \(y = \mathrm { e } ^ { 2 x } \sec 3 x\),
(b) find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\). The curve with equation \(y = \mathrm { e } ^ { 2 x } \sec 3 x , - \frac { \pi } { 6 } < x < \frac { \pi } { 6 }\), has a minimum turning point at \(( a , b )\).
(c) Find the values of the constants \(a\) and \(b\), giving your answers to 3 significant figures.

5. (i) Use the derivatives of \(\sin x\) and \(\cos x\) to prove that $$\frac { \mathrm { d } } { \mathrm {~d} x } ( \tan x ) = \sec ^ { 2 } x$$ The tangent to the curve \(y = 2 x \tan x\) at the point where \(x = \frac { \pi } { 4 }\) meets the \(y\)-axis at the point \(P\).
(ii) Find the \(y\)-coordinate of \(P\) in the form \(k \pi ^ { 2 }\) where \(k\) is a rational constant.

3

1. Show that the derivative of \(\sec x\) can be written as \(\sec x \tan x\).
2. Find \(\int \frac { \tan x } { \sqrt { 1 + \cos 2 x } } \mathrm {~d} x\).

7 Differentiate \(\cos x\) with respect to \(x\), from first principles.

9. Given that \(x\) is measured in radians, prove, from the first principles, that $$\frac { \mathrm { d } } { \mathrm {~d} x } ( \sin x ) = \cos x$$ You may assume the formula for \(\sin ( A \pm B )\) and that as \(h \rightarrow 0 , \frac { \sin h } { h } \rightarrow 1\) and \(\frac { \cos h - 1 } { h } \rightarrow 0\).

12. $$y = \sin x$$ where \(x\) is measured in radians.
Use differentiation from first principles to show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \cos x$$ You may

• use without proof the formula for \(\sin ( A \pm B )\)
• assume that as \(h \rightarrow 0 , \frac { \sin h } { h } \rightarrow 1\) and \(\frac { \cos h - 1 } { h } \rightarrow 0\)

10. Given that \(\theta\) is measured in radians, prove, from first principles, that the derivative of \(\sin \theta\) is \(\cos \theta\) You may assume the formula for \(\sin ( A \pm B )\) and that as \(h \rightarrow 0 , \frac { \sin h } { h } \rightarrow 1\) and \(\frac { \cos h - 1 } { h } \rightarrow 0\)

1. Given that \(\theta\) is measured in radians, prove, from first principles, that

$$\frac { \mathrm { d } } { \mathrm {~d} \theta } ( \cos \theta ) = - \sin \theta$$ You may assume the formula for \(\cos ( A \pm B )\) and that as \(h \rightarrow 0 , \frac { \sin h } { h } \rightarrow 1\) and \(\frac { \cos h - 1 } { h } \rightarrow 0\) (5)

5 Fig. 5.1 shows part of the curve \(y = x ^ { \frac { 1 } { 2 } }\). P is the point \(( 1,1 )\) and \(Q\) is the point on the curve with \(x\)-coordinate \(1 + h\). \begin{figure}[h] \end{figure} Table 5.2 shows, for different values of \(h\), the coordinates of P , the coordinates of Q , the change in \(y\) from P to Q and the gradient of the chord PQ . \begin{table}[h] \end{table}

1. Fill in the missing values for the case \(h = 1\) in the copy of Table 5.2 in the Printed Answer Booklet. Give your answers correct to 6 decimal places where necessary.
2. Explain how the sequence of values in the last column of Table 5.2 relates to the gradient of the curve \(y = x ^ { \frac { 1 } { 2 } }\) at the point \(P\).
3. Use calculus to find the gradient of the curve at the point P .

2 State the value of $$\lim _ { h \rightarrow 0 } \frac { \sin ( \pi + h ) - \sin \pi } { h }$$ Circle your answer. \(\cos h\) -1
0
1

Use the derivatives of \(\sin x\) and \(\cos x\) to prove that the derivative of \(\tan x\) is \(\sec^2 x\). [4]

Jodie is attempting to use differentiation from first principles to prove that the gradient of \(y = \sin x\) is zero when \(x = \frac{\pi}{2}\) Jodie's teacher tells her that she has made mistakes starting in Step 4 of her working. Her working is shown below. \includegraphics{figure_11} Complete Steps 4 and 5 of Jodie's working below, to correct her proof. [4 marks] Step 4 \quad For gradient of curve at A, Step 5 \quad Hence the gradient of the curve at A is given by

A curve has equation \(y = xe^{\frac{x}{2}}\) Show that the curve has a single point of inflection and state the exact coordinates of this point of inflection. [8 marks]