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Ex 17.3
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Solutions
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(a)
\begin{align} {d \over dx} (\ln x + 2x) & = {1 \over x} + 2 \end{align}
(b)
\begin{align} u & = x - 1 &&& v & = \ln x \\ {du \over dx} & = 1 &&& {dv \over dx} & = {1 \over x} \end{align} \begin{align} {d \over dx} [(x - 1)\ln x) ] & = (x - 1)\left(1 \over x\right) + (\ln x)(1) \phantom{000000} [\text{Product rule}] \\ & = {1 \over x}(x - 1) + \ln x \\ & = {x \over x } - {1 \over x} + \ln x \\ & = 1 - {1 \over x} + \ln x \end{align}
(c)
\begin{align} u & = x &&& v & = 1 + \ln x \\ {du \over dx} & = 1 &&& {dv \over dx} & = {1 \over x} \end{align} \begin{align} {d \over dx} \left( x \over 1 + \ln x\right) & = { (1 + \ln x)(1) - (x) \left(1 \over x\right) \over (1 + \ln x)^2 } \phantom{000000} [\text{Quotient rule}] \\ & = {1 + \ln x - {x \over x} \over (1 + \ln x)^2 } \\ & = {1 + \ln x - 1 \over (1 + \ln x)^2 } \\ & = { \ln x \over (1 + \ln x)^2 } \end{align}
(d)
\begin{align} u & = \sin x &&& v & = \ln x \\ {du \over dx} & = \cos x &&& {dv \over dx} & = {1 \over x} \end{align} \begin{align} {d \over dx} \left( \sin x \over \ln x\right) & = {(\ln x)(\cos x) - (\sin x)\left(1 \over x\right) \over (\ln x)^2 } \phantom{000000} [\text{Quotient rule}] \\ & = { \cos x \ln x - {1 \over x} \sin x \over (\ln x)^2 } \\ & = { \cos x \ln x - {1 \over x} \sin x \over (\ln x)^2 } \times {x \over x} \\ & = { x \cos x \ln x - \sin x \over x (\ln x)^2 } \end{align}
(a)
\begin{align} {d \over dx} [\ln (5x + 1)] & = {5(1) \over 5x + 1} \\ & = {5 \over 5x + 1} \end{align}
(b)
\begin{align} {d \over dx} [\ln (4x - 3)^2 ] & = {d \over dx} [2 \ln (4x - 3)] \phantom{000000} [\text{Power law (logarithms)}] \\ & = 2 \left[ 4(1) \over 4x - 3 \right] \\ & = {2 \over 1} \left( 4 \over 4x - 3\right) \\ & = {8 \over 4x - 3} \end{align}
(c)
\begin{align} {d \over dx} [\ln (8 - x^3)] & = {- 3x^2 \over 8 - x^3} \\ & = {3x^2 \over -(8 - x^3)} \\ & = {3x^2 \over x^3 - 8} \end{align}
(d)
\begin{align} u & = x &&& v & = \ln (5 - 2x) \\ {du \over dx} & = 1 &&& {dv \over dx} & = {-2(1) \over 5 - 2x} \\ & &&& & = {- 2 \over 5 - 2x} \\ & &&& & = {2 \over -(5 - 2x)} \\ & &&& & = {2 \over 2x - 5} \end{align} \begin{align} {d \over dx} [ x \ln (5 - 2x) ] & = (x)\left(2 \over 2x - 5\right) + [\ln (5 - 2x)](1) \phantom{000000} [\text{Product rule}] \\ & = {2x \over 2x - 5} + \ln (5 - 2x) \end{align}
(a)
\begin{align} y & = \ln (2x - 5) \\ \\ \text{When } & y = 0, \\ 0 & = \ln (2x - 5) \\ 0 & = \log_e (2x - 5) \\ e^0 & = 2x - 5 \\ 1 & = 2x - 5 \\ 1 + 5 & = 2x \\ 6 & = 2x \\ {6 \over 2} & = x \\ 3 & = x \\ \\ \text{Coordinates} & \text{ of point is } (3, 0) \\ \\ y & = \ln (2x - 5) \\ \\ {dy \over dx} & = {2(1) \over 2x - 5} \\ & = {2 \over 2x - 5} \\ \\ \text{When } & x = 3, \\ {dy \over dx} & = {2 \over 2(3) - 5} \\ & = 2 \end{align}
(b)
\begin{align} y & = \ln (3 - 2x) \\ \\ \text{When } & y = 0, \\ 0 & = \ln (3 - 2x) \\ 0 & = \log_e (3 - 2x) \\ e^0 & = 3 - 2x \\ 1 & = 3 - 2x \\ 2x & = 3 -1 \\ 2x & = 2 \\ x & = {2 \over 2} \\ & = 1 \\ \\ \text{Coordinates} & \text{ of point is } (1, 0) \\ \\ y & = \ln (3 - 2x) \\ \\ {dy \over dx} & = {-2(1) \over 3 - 2x} \\ & = {-2 \over 3 - 2x} \\ \\ \text{When } & x = 1, \\ {dy \over dx} & = {-2 \over 3 - 2(1)} \\ & = -2 \end{align}
(c)
\begin{align} y & = 3 \ln (x - 3) \\ \\ \text{When } & y = 0, \\ 0 & = 3 \ln (x - 3) \\ {0 \over 3} & = \ln (x - 3) \\ 0 & = \log_e (x - 3) \\ e^0 & = x - 3 \\ 1 & = x - 3 \\ 1 + 3 & = x \\ 4 & = x \\ \\ \text{Coordinates} & \text{ of point is } (4, 0) \\ \\ y & = 3 \ln (x - 3) \\ \\ {dy \over dx} & = 3 \left(1 \over x - 3\right) \\ & = {3 \over x - 3} \\ \\ \text{When } & x = 4, \\ {dy \over dx} & = {3 \over (4) - 3} \\ & = 3 \end{align}
(d)
\begin{align} y & = \ln (3x - 2)^2 \\ & = 2 \ln (3x - 2) \phantom{000000} [\text{Power law (logarithms)}] \\ \\ \text{When } & y = 0, \\ 0 & = 2\ln (3x - 2) \\ {0 \over 2} & = \ln (3x - 2) \\ 0 & = \log_e (3x - 2) \\ e^0 & = 3x - 2 \\ 1 & = 3x - 2 \\ 3 & = 3x \\ {3 \over 3} & = x \\ 1 & = x \\ \\ \text{Coordinates} & \text{ of point is } (1, 0) \\ \\ y & = 2 \ln (3x - 2) \\ \\ {dy \over dx} & = 2 \left(3(1) \over 3x - 2\right) \\ & = {6 \over 3x - 2} \\ \\ \text{When } & x = 1, \\ {dy \over dx} & = {6 \over 3(1) - 2} \\ & = 6 \end{align}
Question 4 - Nature of stationary point
(i)
\begin{align} u & = x &&& v & = \ln x \\ {du \over dx} & = 1 &&& {dv \over dx} & = {1 \over x} \end{align} \begin{align} {dy \over dx} & = { (\ln x)(1) - (x)\left(1 \over x\right) \over (\ln x)^2 } \phantom{000000} [\text{Quotient rule}] \\ & = { \ln x - 1 \over (\ln x)^2 } \end{align}
(ii) I chose to use First Derivative Test since the differentiation to obtain ${d^2 y \over dx^2}$ is tedious
\begin{align} {dy \over dx} & = {\ln x - 1 \over (\ln x)^2} \\ \\ \text{When } & {dy \over dx} = 0, \\ 0 & = {\ln x - 1 \over (\ln x)^2} \\ 0 & = \ln x - 1 \\ 1 & = \ln x \\ 1 & = \log_e x \\ e^1 & = x \\ e & = x \\ \\ \text{Substitute } & x = e \text{ into eqn of curve,} \\ y & = {(e) \over \ln (e)} \\ & = {e \over 1} \\ & = e \\ \\ \text{Coordinates} & \text{ of stationary point is } (e, e) \end{align}
$x$ | $2.7$ | $e$ | $2.8$ |
---|---|---|---|
${dy \over dx}$ | $ - $ | $ 0 $ | $ + $ |
Slope | \ | - | / |
$$ \therefore (e, e) \text{ is a minimum point} $$
Question 5 - Increasing function
\begin{align} u & = \ln (x + 1) &&& v & = x + 1 \\ {du \over dx} & = {1 \over x + 1} &&& {dv \over dx} & = 1 \end{align} \begin{align} {dy \over dx} & = {(x + 1)\left(1 \over x + 1\right) - [\ln (x + 1)](1) \over (x + 1)^2 } \phantom{000000} [\text{Quotient rule}] \\ & = { 1 - \ln(x + 1) \over (x + 1)^2 } \\ \\ {dy \over dx} & > 0 \phantom{000000} [\text{Increasing function}] \\ {1 - \ln (x + 1) \over (x + 1)^2} & > 0 \\ \\ \text{For } x > -1, & \phantom{.} (x + 1)^2 > 0 \\ \\ \implies 1 - \ln (x + 1) & > 0 \\ - \ln (x + 1) & > - 1 \\ \ln (x + 1) & < 1 \\ \log_e (x + 1) & < 1 \\ x + 1 & < e^1 \\ x + 1 & < e \\ x & < e - 1 \\ \\ \text{Since } x > -1, & \phantom{..} -1 < x < e - 1 \end{align}
Question 6 - Show that function is increasing
\begin{align} C & = 11.9 + 0.1t + \ln (t + 1) \\ \\ {d C \over dt} & = 0.1(1) + {1 \over t + 1} \\ & = 0.1 + {1 \over t + 1} \\ \\ \text{For } 0 & \phantom{.} \le t \le 10, \phantom{..} t + 1 > 0 \end{align} \begin{align} \implies {1 \over t + 1} & > 0 \\ 0.1 + {1 \over t + 1} & > 0 \\ {dC \over dt} & > 0 \\ \\ \therefore C \text{ is increasing} & \text{ for } 0 \le t \le 10 \end{align}
(a)
\begin{align} u & = 3x &&& v & = \ln \sqrt{x^2 - 2x - 3} \\ & &&& & = \ln (x^2 - 2x - 3)^{1 \over 2} \\ & &&& & = {1 \over 2} \ln (x^2 - 2x - 3) \phantom{000000} [\text{Power law}] \\ {du \over dx} & = 3 &&& {dv \over dx} & = {1 \over 2} \left(2x - 2 \over x^2 - 2x - 3\right) \\ & &&& & = {2(x - 1) \over 2(x^2 - 2x - 3) } \\ & &&& & = {x - 1 \over x^2 - 2x - 3} \end{align} \begin{align} {d \over dx}( 3x \ln \sqrt{x^2 - 2x - 3} ) & = (3x)\left(x - 1 \over x^2 - 2x - 3\right) + \left(\ln \sqrt{x^2 - 2x - 3}\right) (3) \phantom{000000} [\text{Product rule}] \\ & = {3x \over 1} \left(x - 1 \over x^2 - 2x - 3\right) + 3 \ln \sqrt{x^2 - 2x - 3} \\ & = {3x(x - 1) \over x^2 - 2x - 3} + 3 \ln \sqrt{x^2 - 2x - 3} \end{align}
(b)
\begin{align} {d \over dx} [ \ln ( \sin x \cos x) ] & = {d \over dx} [ \ln (\sin x) + \ln (\cos x) ] \phantom{000000} [\text{Product law (logarithms)}] \\ & = {\cos x \over \sin x} + {- \sin x \over \cos x} \\ & = \cot x - \tan x \\ & = {1 \over \tan x} - \tan x \\ & = {1 \over \tan x} - {\tan^2 x \over \tan x} \\ & = {1 - \tan^2 x \over \tan x} \\ & = 2 \left(1 - \tan^2 x \over 2\tan x\right) \\ & = 2 \left(1 \over \tan 2x\right) \phantom{000000} [\text{Double angle formula}]\\ & = 2 \cot 2x \end{align}
(c)
\begin{align} {d \over dx} \left[ \ln \left( 3 + e^{-{1 \over 2}x} \right) \right] & = { \left(-{1 \over 2}\right) e^{-{1 \over 2}x} \over 3 + e^{-{1 \over 2}x} } \\ & = - { e^{-{1 \over 2}x} \over 2 \left( 3 + e^{-{1 \over 2}x} \right) } \end{align}
(d)
\begin{align} {d \over dx} \left[ \ln \left( \sqrt{x^2 + 1} - x \right) \right] & = {d \over dx} \left\{ \ln \left[ (x^2 + 1)^{1 \over 2} - x \right] \right\} \\ & = { {1 \over 2}(x^2 + 1)^{-{1 \over 2}} . (2x) - 1 \over (x^2 + 1)^{1 \over 2} - x } \\ & = { x (x^2 + 1)^{-{1 \over 2}} - 1 \over \sqrt{x^2 + 1} - x } \\ & = { {x \over \sqrt{x^2 + 1}} - 1 \over \sqrt{x^2 + 1} - x } \\ & = { {x \over \sqrt{x^2 + 1}} - {\sqrt{x^2 + 1} \over \sqrt{x^2 + 1}} \over \sqrt{x^2 + 1} - x} \\ & = { {x - \sqrt{x^2 + 1} \over \sqrt{x^2 + 1}} \over \sqrt{x^2 + 1} - x} \\ & = {x - \sqrt{x^2 + 1} \over \sqrt{x^2 + 1}} \div \left( \sqrt{x^2 + 1} - x \right) \\ & = {x - \sqrt{x^2 + 1} \over \sqrt{x^2 + 1}} \times {1 \over \sqrt{x^2 + 1} - x } \\ & = {- (\sqrt{x^2 + 1} - x) \over \sqrt{x^2 + 1}} \times {1 \over \sqrt{x^2 + 1} - x } \\ & = { -1 \over \sqrt{x^2 + 1}} \end{align}
(e) Note $e^2$ is a constant
\begin{align} \text{Let } y & = \ln \left[ e^2 (3x - 2) \over 4 - x \right] \\ & = \ln [e^2 (3x - 2)] - \ln (4 - x) \phantom{0000000} [\text{Quotient law}] \\ & = \ln (3e^2 x - 2e^2) - \ln (4 - x) \\ \\ {dy \over dx} & = { 3e^2 (1) \over 3e^2 x - 2e^2} - { -1 \over 4 - x} \\ & = {3 e^2 \over e^2 (3x - 2) } + {1 \over 4 - x} \\ & = {3 \over 3x - 2} + {1 \over 4 - x} \end{align}
(f)
\begin{align} \text{Let } y & = \ln \left[ (x + 2)^3 \cos 3x \over \sin x \right] \\ & = \ln \left[ (x + 2)^3 \cos 3x \right] - \ln (\sin x) \phantom{0000000} [\text{Quotient law}] \\ & = \ln (x + 2)^3 + \ln (\cos 3x) - \ln (\sin x) \phantom{000.} [\text{Product law}] \\ & = 3 \ln (x + 2) + \ln (\cos 3x) - \ln (\sin x) \phantom{000} [\text{Power law}] \\ \\ {dy \over dx} & = 3 \left(1 \over x + 2\right) + {(3)(-\sin 3x) \over \cos 3x} - {\cos x \over \sin x} \\ & = {3 \over 1} \left(1 \over x + 2\right) - 3 \tan 3x - \cot x \\ & = {3 \over x + 2} - 3 \tan 3x - \cot x \end{align}
(a)
\begin{align} y & = \ln (2x - 1) \\ \\ {dy \over dx} & = {2(1) \over 2x - 1} \\ & = {2 \over 2x - 1} \\ \\ \text{For all } & \text{real values of } x, \\ {2 \over 2x - 1} & \ne 0 \\ \implies {dy \over dx} & \ne 0 \\ \\ \therefore \text{Curve has} & \text{ no stationary points} \end{align}
(b) The coordinates of the x-intercept is (2, 0)
\begin{align} y & = \ln (3x - k) \\ \\ \text{Using } & (2, 0), \\ 0 & = \ln [3(2) - k] \\ 0 & = \log_e (6 - k) \\ e^0 & = 6 - k \\ 1 & = 6 - k \\ k & = 6 - 1 \\ & = 5 \\ \\ \text{Eqn of curve: } & y = \ln (3x - 5) \\ \\ {dy \over dx} & = {3(1) \over 3x - 5} \\ & = {3 \over 3x - 5} \\ \\ \text{When } & x = 2, \\ {dy \over dx} & = {3 \over 3(2) - 5} \\ & = 3 \\ \\ \text{Gradient of normal} & = -{1 \over 3} \\ \\ y & = mx + c \\ y & = -{1 \over 3}x + c \\ \\ \text{Using } & (2, 0), \\ 0 & = -{1 \over 3}(2) + c \\ 0 & = -{2 \over 3} + c \\ {2 \over 3} & = c \\ \\ \text{Eqn of normal: } & y = -{1 \over 3}x + {2 \over 3} \\ 3y & = - x + 2 \\ 3y + x & = 2 \end{align}
Question 9 - Connected rate of change
(i)
\begin{align} y & = \ln (x^2 - 4) \\ \\ {dy \over dx} & = {2x \over x^2 - 4} \end{align}
(ii)
\begin{align} {dx \over dt} & = p \text{ units per second} \\ \\ {dy \over dt} & = {dy \over dx} \times {dx \over dt} \\ & = {2x \over x^2 - 4} \times p \\ & = {2x \over x^2 - 4} \times {p \over 1} \\ & = {2p x \over x^2 - 4} \\ \\ \text{When } & x = 3, \\ {dy \over dt} & = {2p (3) \over (3)^2 - 4} \\ & = {6p \over 5} \\ & = {6 \over 5}p \text{ units per second} \end{align}
Question 10 - Connected rate of change
\begin{align} {dx \over dt} & = 3 \text{ units per second} \\ \\ y & = \ln (2 + e^{-x} ) \\ \\ {dy \over dx} & = { (-1)e^{-x} \over 2 + e^{-x} } \\ & = { - e^{-x} \over 2 + e^{-x} } \\ \\ {dy \over dt} & = {dy \over dx} \times {dx \over dt} \\ & = { - e^{-x} \over 2 + e^{-x} } \times 3 \\ & = { - e^{-x} \over 2 + e^{-x} } \times {3 \over 1} \\ & = { - 3 e^{-x} \over 2 + e^{-x} } \\ \\ \text{When } & x = 1, \\ {dy \over dt} & = { - 3e^{-1} \over 2 + e^{-1} } \\ & = { - 3e^{-1} \over 2 + e^{-1} } \times {e \over e} \\ & = { - 3 e^{-1 + 1} \over 2e + e^{-1 + 1} } \\ & = { - 3 e^0 \over 2e + e^0 } \\ & = { - 3(1) \over 2e + 1 }\\ & = {- 3 \over 2e + 1} \text{ units per second} \end{align}
Question 11 - Minimum point of the curve
(i)
\begin{align} u & = 2x &&& v & = \ln x \\ {du \over dx} & = 2 &&& {dv \over dx} & = {1 \over x} \end{align} \begin{align} {dy \over dx} & = (2x)\left(1 \over x\right) + (\ln x)(2) \phantom{000000} [\text{Product rule}] \\ & = {2x \over 1}\left(1 \over x\right) + 2 \ln x \\ & = {2x \over x} + 2 \ln x \\ & = 2 + 2 \ln x \\ & = 2(1 + \ln x) \end{align}
(ii)
\begin{align} {dy \over dx} & = 2(1 + \ln x) \\ \\ \text{Let } & {dy \over dx} = 0, \\ 0 & = 2(1 + \ln x) \\ {0 \over 2} & = 1 + \ln x \\ 0 & = 1 + \ln x \\ - \ln x & = 1 \\ \ln x & = -1 \\ \log_e x & = -1 \\ x & = e^{-1} \\ & = {1 \over e} \end{align}
(iii)
\begin{align} {dy \over dx} & = 2(1 + \ln x) \\ & = 2 + 2\ln x \\ \\ {d^2 y \over dx^2} & = 2 \left(1 \over x\right) \\ & = {2 \over 1} \left(1 \over x\right) \\ & = {2 \over x} \\ \\ \text{When } & x = {1 \over e}, \\ {d^2 y \over dx^2} & = {2 \over {1 \over e}} \\ & = 2 \div {1 \over e} \\ & = 2 \times e \\ & = 2e \end{align}
Question 12 - Maximum point of the curve
(i)
\begin{align} u & = \ln x &&& v & = x^2 \\ {du \over dx} & = {1 \over x} &&& {dv \over dx} & = 2x \end{align} \begin{align} {dy \over dx} & = {(x^2) \left(1 \over x\right) - (\ln x)(2x) \over (x^2)^2 } \phantom{000000} [\text{Quotient rule}] \\ & = {x - 2x \ln x \over x^4} \\ & = { x (1 - 2 \ln x) \over x^4 } \\ & = { 1 - 2\ln x \over x^3 } \end{align}
(ii)
\begin{align} {dy \over dx} & = { 1 - 2\ln x \over x^3 } \\ \\ \text{Let } & {dy \over dx} = 0, \\ 0 & = {1 - 2 \ln x \over x^3} \\ 0 & = 1 - 2 \ln x \\ 2 \ln x & = 1 \\ \ln x & = {1 \over 2} \\ \log_e x & = {1 \over 2} \\ x & = e^{1 \over 2} \\ & = \sqrt{e} \end{align}
(iii)
\begin{align}
{dy \over dx} & = { 1 - 2\ln x \over x^3 }
\end{align}
\begin{align}
u & = 1 - 2 \ln x &&& v & = x^3 \\
{du \over dx} & = - 2 \left(1 \over x\right) &&& {dv \over dx} & = 3x^2 \\
& = -{2 \over x}
\end{align}
\begin{align}
{d^2 y \over dx^2} & = { (x^3) \left(-{2 \over x}\right) - (1 - 2 \ln x)(3x^2)
\over (x^3)^2 } \phantom{000000} [\text{Quotient rule}] \\
& = { {-2x^3 \over x} - 3x^2 (1 - 2 \ln x) \over x^6 } \\
& = { - 2x^2 - 3x^2 (1 - 2 \ln x) \over x^6 } \\
& = { x^2 [ - 2 - 3(1 - 2 \ln x) ] \over x^6 } \\
& = { - 2 - 3(1 - 2 \ln x) \over x^4 } \\
& = { - 2 - 3 + 6 \ln x \over x^4 } \\
& = { - 5 + 6 \ln x \over x^4 } \\
\\
\text{When } & x = e^{1 \over 2}, \\
{d^2 y \over dx^2} & = { - 5 + 6 \ln e^{1 \over 2} \over (e^{1 \over 2})^4 } \\
& = { - 5 + 6 \left(1 \over 2\right) \over e^2 } \\
& = { - 2 \over e^2}
\end{align}
Question 13 - Maximum speed of signal
\begin{align}
V & = \lambda x^2 \ln \left(1 \over x\right) \\
& = \lambda x^2 \ln x^{-1} \\
& = \lambda x^2 (-1) \ln x \phantom{00000000} [\text{Power law (logarithms)}] \\
& = -\lambda x^2 \ln x
\end{align}
\begin{align}
u & = - \lambda x^2 &&& v & = \ln x \\
{du \over dx} & = - \lambda (2)x &&& {dv \over dx} & = {1 \over x} \\
& = - 2\lambda x
\end{align}
\begin{align}
{dV \over dx} & = (-\lambda x^2)\left(1 \over x\right) + (\ln x)(- 2 \lambda x)
\phantom{000000} [\text{Product rule}] \\
& = { -\lambda x^2 \over x } - 2 \lambda x \ln x \\
& = -\lambda x - 2\lambda x \ln x
\end{align}
\begin{align}
u & = - 2 \lambda x &&& v & = \ln x \\
{du \over dx} & = -2\lambda (1) &&& {dv \over dx} & = {1 \over x} \\
& = - 2 \lambda
\end{align}
\begin{align}
{d^2 V \over dx^2} & = - \lambda(1) + (-2 \lambda x)\left(1 \over x\right) + (\ln x)(-2 \lambda) \\
& = - \lambda + {-2\lambda x \over x} - 2 \lambda \ln x \\
& = - \lambda - 2 \lambda - 2 \lambda \ln x \\
& = - 3 \lambda - 2 \lambda \ln x
\end{align}
\begin{align}
\text{Let } & {dV \over dx} = 0, \\
0 & = - \lambda x - 2\lambda x \ln x \\
0 & = - \lambda x (1 + 2 \ln x)
\end{align}
\begin{align}
- \lambda x & = 0 && \text{ or } & 1 + 2 \ln x & = 0 \\
x & = 0 \text{ (Reject, since } 0 < x < 1) &&& 2 \ln x & = -1 \\
& &&& \ln x & = -{1 \over 2} \\
& &&& \log_e x & = -{1 \over 2} \\
& &&& x & = e^{-{1 \over 2}} \\
& &&& & = {1 \over \sqrt{e}} \\
\\
& &&& \text{Substitute } & \text{into } {d^2 V \over dx^2}, \\
& &&& {d^2 V \over dx^2} & = - 3 \lambda - 2\lambda \ln e^{-{1 \over 2}} \\
& &&& & = -3 \lambda - 2 \lambda \left(-{1 \over 2}\right) \\
& &&& & = -3 \lambda + \lambda \\
& &&& & = -2 \lambda
\end{align}
\begin{align}
\text{Since } \lambda > 0, & \phantom{.} {d^2 V \over dx^2} > 0 \\
\\
\therefore \text{When } x = {1 \over \sqrt{e}}, & \phantom{.} \text{ speed (} V)\text{ is a maximum}
\end{align}
Note: I don't think this will be tested in O levels (more for A levels)
\begin{align} y & = 2^x \\ \\ \text{Take } \ln & \text{ of both sides,} \\ \ln y & = \ln 2^x \\ \ln y & = x \ln 2 \phantom{000000} [\text{Power law (logarithms)}] \\ \ln y & = (\ln 2) x \\ \\ \text{Differentiate } & \text{both sides w.r.t } x, \\ {d \over dx} ( \ln y) & = {d \over dx} [ (\ln 2) x ] \\ {1 \over y} . {dy \over dx} & = (\ln 2)(1) \\ {1 \over y} {dy \over dx} & = \ln 2 \\ {dy \over dx} & = (\ln 2) y \phantom{00} \text{ (Shown)} \end{align}
Note: I don't think this will be tested in O levels (more for A levels)
\begin{align} y & = (2 + x^2)^3 (1 - x^3)^4 \\ \\ \text{Take } \ln & \text{ of both sides,} \\ \ln y & = \ln \left[ (2 + x^2)^3 (1 - x^3)^4 \right] \\ \ln y & = \ln (2 + x^2)^3 + \ln (1 - x^3)^4 \phantom{000000} [\text{Product law (logarithms)}] \\ \ln y & = 3\ln (2 + x^2) + 4 \ln (1 - x^3) \phantom{00000} [\text{Power law (logarithms)}] \\ \\ \text{Differentiate } & \text{both sides w.r.t } x, \\ {d \over dx} ( \ln y) & = {d \over dx} \left[ 3\ln (2 + x^2) + 4 \ln (1 - x^3) \right] \\ {1 \over y} . {dy \over dx} & = 3 \left(2x \over 2 + x^2\right) + 4 \left(- 3x^2 \over 1 - x^3\right) \\ {1 \over y} . {dy \over dx} & = {6x \over 2 + x^2} - {12x^2 \over 1 - x^3} \\ {dy \over dx} & = y \left( {6x \over 2 + x^2} - {12x^2 \over 1 - x^3} \right) \\ & = \left[ (2 + x^2)^3 (1 - x^3)^4 \right] \left( {6x \over 2 + x^2} - {12x^2 \over 1 - x^3} \right) \\ & = 6x(2 + x^2)^2 (1 - x^3)^4 - 12x^2 (2 + x^2)^3 (1 - x^3)^3 \\ & = 6x(2 + x^2)^2 (1 - x^3)^3 [ (1 - x^3) - 2x (2 + x^2) ] \\ & = 6x(2 + x^2)^2 (1 - x^3)^3 (1 - x^3 - 4x - 2x^3) \\ & = 6x(2 + x^2)^2 (1 - x^3)^3 (1 - 4x - 3x^3) \end{align}
Note: I don't think this will be tested in O levels (more for A levels)
(i)
\begin{align} y & = { \left(x + {1 \over x}\right)^5 (5 + x) e^x \over 10^x \ln x } \\ \ln y & = \ln \left[ \left(x + {1 \over x}\right)^5 (5 + x) e^x \over 10^x \ln x \right] \\ \ln y & = \ln \left[ \left(x + {1 \over x}\right)^5 (5 + x) e^x \right] - \ln (10^x \ln x ) \\ \ln y & = \ln \left(x + {1 \over x}\right)^5 + \ln (5 + x) + \ln e^x - \ln 10^x - \ln (\ln x) \\ \ln y & = 5 \ln (x + x^{-1}) + \ln (5 + x) + x - x \ln 10 - \ln (\ln x) \\ \\ {1 \over y} {dy \over dx} & = 5 \left(1 - x^{-2} \over x + x^{-1} \right) + {1 \over 5 + x} + 1 - \ln 10 - { {1 \over x} \over \ln x } \\ {1 \over y} {dy \over dx} & = 5 \left( 1 - {1 \over x^2} \over x + {1 \over x} \right) + {1 \over 5 + x} + 1 - \ln 10 - { 1 \over x \ln x} \\ {1 \over y} {dy \over dx} & = {5 \left(1 - {1 \over x^2} \right) \over x + {1 \over x} } + {1 \over 5 + x} + 1 - \ln 10 - {1 \over x \ln x} \\ \\ {dy \over dx} & = y\left[ {5 \left(1 - {1 \over x^2} \right) \over x + {1 \over x} } + {1 \over 5 + x} + 1 - \ln 10 - {1 \over x \ln x} \right] \\ \end{align}
(ii)
\begin{align} y & = { (\cos^5 x) (x^2 + 5) (\ln x) \over e^{x + 1} } \\ \ln y & = \ln \left[ (\cos^5 x) (x^2 + 5) (\ln x) \over e^{x + 1} \right] \\ \ln y & = \ln [(\cos^5 x)(x^2 + 5)(\ln x)] - \ln e^{x + 1} \\ \ln y & = \ln (\cos^5 x) + \ln (x^2 + 5) + \ln (\ln x) - (x + 1) \\ \ln y & = 5 \ln (\cos x) + \ln (x^2 + 5) + \ln (\ln x) - x - 1 \\ \\ {1 \over y} {dy \over dx} & = 5 \left(- \sin x \over \cos x\right) + {2x \over x^2 + 5} + { {1 \over x} \over \ln x } - 1 \\ {1 \over y} {dy \over dx} & = - 5 \tan x + {2x \over x^2 + 5} + {1 \over x \ln x} - 1 \\ {dy \over dx} & = y \left( - 5 \tan x + {2x \over x^2 + 5} + {1 \over x \ln x} - 1 \right) \end{align}
(i)
\begin{align} y & = ax - x \ln x \\ \\ \text{Let } & y = 0, \\ 0 & = ax - x \ln x \\ 0 & = x (a - \ln x) \end{align} \begin{align} x & = 0 \text{ (Reject, since } x > 0) && \text{ or } & a - \ln x & = 0 \\ & &&& - \ln x & = -a \\ & &&& \ln x & = a \\ & &&& \log_e x & = a \\ & &&& x & = e^a \end{align}
(ii)
\begin{align} u & = x &&& v & = \ln x \\ {du \over dx} & = 1 &&& {dv \over dx} & = {1 \over x} \end{align} \begin{align} {dy \over dx} & = a(1) - \left[ (x) \left(1 \over x\right) + (\ln x)(1) \right] \\ & = a - (1 + \ln x) \\ & = a - 1 - \ln x \\ \\ {d^2 y \over dx^2 } & = - {1 \over x} \\ \\ \text{Let } & {dy \over dx} = 0, \\ 0 & = a - 1 - \ln x \\ \ln x & = a -1 \\ \log_e x & = a - 1 \\ x & = e^{a - 1} \\ \\ \text{Substitute } & x = e^{a - 1} \text{ into eqn of curve,} \\ y & = a(e^{a - 1}) - (e^{a - 1}) \ln e^{a - 1} \\ & = a e^{a - 1} - e^{a - 1} (a - 1) \ln e \\ & = a e^{a - 1} - e^{a - 1} (a - 1) (1) \\ & = a e^{a - 1} - a e^{a - 1} + e^{a - 1} \\ & = e^{a - 1} \\ \\ \text{Coordinates } & \text{of stationary point is } (e^{a - 1}, e^{a - 1}) \\ \\ \text{Substitute } & x = e^{a - 1} \text{ into } {d^2 y \over dx^2} ,\\ {d^2 y \over dx^2 } & = - {1 \over e^{a - 1}} \\ \\ \text{Since } a > 0 & \text{ and } e^{a - 1} > 0, \phantom{.} {d^2 y \over dx^2} < 0 \\ \\ \therefore (e^{a - 1}, e^{a - 1}) & \text{ is a maximum point} \end{align}
(i) The dotted line denotes the portion of y = x ln x that was reflected about the x-axis to obtain the graph of y = |x ln x|
(ii)
\begin{align} \text{Consider } y & = x \ln x \\ \\ u & = x \phantom{00000000} {du \over dx} = 1 \\ v & = \ln x \phantom{000000} {dv \over dx} = {1 \over x} \\ \\ {dy \over dx} & = x \left(1 \over x\right) + (\ln x)(1) \\ & = 1 + \ln x \\ \\ \text{Let } & {dy \over dx} = 0, \\ 0 & = 1 + \ln x \\ - \ln x & = 1 \\ \ln x & = -1 \\ \log_e x & = -1 \\ x & = e^{-1} \\ \\ \text{Substitute } & \text{into } y = x \ln x, \\ y & = e^{-1} \ln e^{-1} \\ y & = e^{-1} (-1) \\ y & = -e^{-1} \\ \\ \text{Minimum } & \text{point on } y = x \ln x : \left({1 \over e}, -{1 \over e} \right) \\ \\ \\ \text{Maximum } & \text{point on } y = |x \ln x| : \left({1 \over e}, {1 \over e} \right) \end{align}
\begin{align} \text{Tom: } {d \over dx}(x^n) & = nx^{n - 1} \\ \\ n \text{ is an integer, } & \text{not a variable} \\ \\ \\ \text{Doris: } {d \over dx} (a^x) & = a^x \ln a \\ \\ a \text{ is an integer, } & \text{not a variable} \\ \\ \\ \text{Let } y & = x^x \\ \ln y & = \ln x^x \\ \ln y & = x \ln x \\ {1 \over y}{dy \over dx} & = x\left(1 \over x\right) + (\ln x)(1) \\ {1 \over y}{dy \over dx} & = 1 + \ln x \\ {dy \over dx} & = y(1 + \ln x) \\ {dy \over dx} & = x^x (1 + \ln x) \end{align}
\begin{align} \text{Curve 1: } y & = \ln x \\ \\ \\ \text{Curve 2: } y & = \ln 3x \\ y & = \ln 3 + \ln x \\ y & = \ln x + \ln 3 \\ y & \approx \ln x + 1.10 \\ \\ \\ \text{Curve 2 is a } & \text{translation of curve 1 by approx. 1.10 units} \end{align}
\begin{align} {d \over dx} [ \ln |ax|] & = {d \over dx} [ \ln (|a||x|)] \\ & = {d \over dx} [ \ln |a| + \ln |x| ] \\ & = 0 + {d \over dx} [\ln |x|] \end{align}