Question
Let \(f(x) = p{x^3} + p{x^2} + qx\).
a.Find \(f'(x)\).[2]
b.Given that \(f'(x) \geqslant 0\), show that \({p^2} \leqslant 3pq\).[5]
▶️Answer/Explanation
Markscheme
\(f'(x) = 3p{x^2} + 2px + q\) A2 N2
Note: Award A1 if only 1 error.
[2 marks]
evidence of discriminant (must be seen explicitly, not in quadratic formula) (M1)
eg \({b^2} – 4ac\)
correct substitution into discriminant (may be seen in inequality) A1
eg \({(2p)^2} – 4 \times 3p \times q,{\text{ }}4{p^2} – 12pq\)
\(f'(x) \geqslant 0\) then \(f’\) has two equal roots or no roots (R1)
recognizing discriminant less or equal than zero R1
eg \(\Delta \leqslant 0,{\text{ }}4{p^2} – 12pq \leqslant 0\)
correct working that clearly leads to the required answer A1
eg \({p^2} – 3pq \leqslant 0,{\text{ }}4{p^2} \leqslant 12pq\)
\({p^2} \leqslant 3pq\) AG N0
[5 marks]
Question
If \(y=\ln (2x-1)\), find \(\frac{\mathrm{d^{2}} y}{\mathrm{d} x^{2}}\).
▶️Answer/Explanation
Ans
\(y=\ln (2x-1)\Rightarrow \frac{\mathrm{d} y}{\mathrm{d} x}=\frac{2}{2x-1}\Rightarrow \frac{\mathrm{d} y}{\mathrm{d} x}=2(2x-1)^{-1}\)
\(\Rightarrow \frac{\mathrm{d^{2}} y}{\mathrm{d} x^{2}}=-2(2x-1)^{-2}(2)\)
\(\Rightarrow \frac{\mathrm{d^{2}} y}{\mathrm{d} x^{2}}=\frac{-4}{(2x-1)^{2}}\)
Question
Let \(y=\sin (kx)-kx\cos (kx)\), where \(k\) is a constant. Show that \(\frac{\mathrm{d} y}{\mathrm{d} x}=k^{2}x\sin (kx)\)
▶️Answer/Explanation
Ans
\(y=\sin (kx)-kx\cos (kx)\)
\(\frac{\mathrm{d} y}{\mathrm{d} x}=k\cos(kx)-k\left \{\cos (kx)+x[-k\ sin (kx)] \right \}\)
\(=k \cos (kx)-k \cos(kx)+k^{2}x\ sin (kx)=k^{2}x\ sin (kx)\)
Question
Consider the function \(f(x)=\ln \sqrt{x^{2}+4}\). Find (a) \(f'(x)\) (b) \({f}”(x)\)
▶️Answer/Explanation
Ans
(a) METHOD 1
\(f(x)=\frac{1}{2}\ln (x^{2}+4)\)
\(f'(x)=\frac{1}{2}\left ( \frac{2x}{x^{2}+4} \right )\)
\(f'(x)=\frac{x}{x^{2}+4}\)
METHOD 2
\(f'(x)=\frac{1}{\sqrt{x^{2}+4}}\times \frac{1}{2}\left ( x^{2}+4 \right )^{-\frac{1}{2}}(2x)\)
\(=\frac{x}{x^{2}+4}\)
(b) \({f}”(x)=\frac{x^{2}+4-x(2x)}{(x^{2}+4)^{2}}\)
\(=\frac{4-x^{2}}{(x^{2}+4)^{2}}\)
Question
Consider the function \(f(t)=3\sec ^{2}t+5t\). Find (a) \(f'(t)\). (b) (ⅰ) \(f(\pi )\); (ⅱ) \(f'(\pi )\);
▶️Answer/Explanation
Ans
(a) METHOD 1
\(f(t)=3\sec ^{2}t+5t=3(\cos t)^{-2}+5t\)
\(f(t)=-6(\cos t)^{-3}(-\sin t)+5=\frac{6\sin t}{cos^{3}t}+5\)
METHOD 2
\(f'(t)=3\times 2\sec t(\sec t\tan t)+5\)
\(=6\sec ^{2}t \tan t+5(=6\tan ^{2}t+6 \tan t+5)\)
(b) \(f(\pi )=\frac{3}{(\cos \pi )^{2}}+5\pi =3+5\pi\)
\(f'(\pi )=\frac{6\sin \pi }{(\cos \pi )^{3}}+5=5\)
Question
The function \(f\) is defined by \(f:x\mapsto 3^{x}\). Find the solution of the equation \({f}”(x)=2\).
▶️Answer/Explanation
Ans
\(f(x)=3^{x}\Rightarrow f(x)=3^{x}\ln 3\)
\(\Rightarrow f'(x)=3^{x}(\ln 3)^{2}\)
\(3^{x}(\ln 3)^{2}=2\)
\(3^{x}=\frac{2}{(\ln 3)^{2}}\)
\(x\ln 3=\ln \left ( \frac{2}{(\ln 3)^{2}} \right )\)
\(x=\frac{\ln \left ( \frac{2}{(\ln 3)^{2}} \right )}{\ln 3}\)
\(=0.460\)
Question
Let \(y=e^{3x}\sin (\pi x)\). Find (a) \(\frac{\mathrm{d} y}{\mathrm{d} x}\). (b) the smallest positive value of \(x\) for which \(\frac{\mathrm{d} y}{\mathrm{d} x}=0\)
▶️Answer/Explanation
Ans
\(y=e^{3x}\sin (\pi x)\)
(a) \(\frac{\mathrm{d} y}{\mathrm{d} x}=3e^{3x}\sin (\pi x)+\pi e^{3x}\cos (\pi x)\)
(b) \(0=e^{3x}(3\sin(\pi x)+\pi \cos (\pi x))\)
\(tan (\pi x)=-\frac{\pi }{3}\)
\(\pi x=-0.80845+\pi\)
\(x=0.7426… (0.743 to 3s.f.)\)