Exercise 13

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Differentiate:

Question 1(i) \[x^{2}\sqrt{4-x^{2}}+\frac{2}{3}\left(4-x^{2}\right)^{\frac{3}{2}}\] Solution
Question 1(ii) \[\sqrt[3]{\left(3x+2\right)^{4}}\]
Question 1(iii) \[\sqrt{\frac{(x-1)(x-2)}{x-3}}\]
Question 1(iv) \[\frac{1+x^{2}}{\sqrt[3]{1-2x}}\]
Question 1(v) \[\frac{\sqrt{x}}{1-x}\]
Question 1(vi) \[\frac{\sqrt{2x^{2}+3}}{5x}\]
Question 1(vii) \[\sqrt{\left(2x^{2}-3\right)\left(4-x^{2}\right)}\]
Question 1(viii) \[x \cdot \left(3x^{2}-7\right)^{\frac{2}{3}}\]

Find [math]\frac{dv}{dp}[/math] when [math]pv^{\frac{2}{3}}=k[/math].

Find the equation of the tangent to the curve [math]\sqrt{x}+\sqrt{y}=5[/math] at the point [math]\left(9, 4\right)[/math].

Find:

Question 4(i) \[\lim_{x \to 0} \frac{\sqrt{1+x}-1}{x}\]
Question 4(ii) \[\lim_{x \to 0} \frac{\sqrt{1+x}-\sqrt{1-x}}{x}\]
Question 4(iii) \[\lim_{x \to \infty} \frac{x}{\sqrt{2x^{2}+1}}\]

Find:

Question 6(i) \[\int \! {\sqrt{x}} \,\mathrm{d}x\]
Question 6(ii) \[\int \! {\sqrt{x^{3}}} \,\mathrm{d}x\]
Question 6(iii) \[\int \! {\frac{1}{\sqrt{x^{3}}}} \,\mathrm{d}x\]
Question 6(iv) \[\int \! {\frac{1}{\sqrt{x}}} \,\mathrm{d}x\]
Question 6(v) \[\int \! {\sqrt[3]{x^{2}}} \,\mathrm{d}x\]
Question 6(vi) \[\int \! {\sqrt{x} \left( x+2 \right)} \,\mathrm{d}x\]
Question 6(vii) \[\int \! {\frac{x^{2}+x+1}{\sqrt{x}}} \,\mathrm{d}x\]
Question 6(viii) \[\int \! {x^{\frac{2}{3}} \left( 1-x \right)} \,\mathrm{d}t\]
Question 6(ix) \[\int \! {\left( \sqrt{x}+\frac{1}{\sqrt{x}} \right)} \,\mathrm{d}x\]
Question 6(x) \[\int \! {\sqrt{3x}} \,\mathrm{d}x\]
Question 6(xi) \[\int \! {\sqrt{3x+1}} \,\mathrm{d}x\]
Question 6(xii) \[\int \! {\sqrt{3-x}} \,\mathrm{d}x\]
Question 6(xiii) \[\int \! {\frac{1}{\sqrt{3-5x}}} \,\mathrm{d}x\]
Question 6(xiv) \[\int \! {\sqrt[3]{\left(3-5x\right)^{2}}} \,\mathrm{d}x\]

Find:

Question 8(i) \[\int \! {x \sqrt{2x^{2}+4}} \,\mathrm{d}x\]
Question 8(ii) \[\int \! {x^{2} \left(2x^{3}-1\right)^{\frac{3}{2}}} \,\mathrm{d}x\]
Question 8(iii) \[\int \! {t \sqrt{t^{2}-1}} \,\mathrm{d}t\]
Question 8(iv) \[\int \! {z \left( 1 - 4x^{2} \right)^{\frac{2}{3}}} \,\mathrm{d}z\]
Question 8(v) \[\int \! {\frac{x^{3}}{\sqrt{1-4x^{2}}}} \,\mathrm{d}x\]
Question 8(vi) \[\int \! {\frac{x^{2}}{\sqrt{2-x^{3}}}} \,\mathrm{d}x\]
Question 8(vii) \[\int \! {\frac{x-1}{\sqrt{x^{2}-2x}}} \,\mathrm{d}x\]
Question 8(viii) \[\int \! {\left(2t-1\right) \sqrt{t^{2}-t}} \,\mathrm{d}t\]
Question 8(ix) \[\int \! {\frac{x}{\sqrt{1-2x^{2}}}} \,\mathrm{d}x\]
Question 8(x) \[\int \! {\frac{\left(x^{\frac{2}{3}}+1\right)^{2}}{\sqrt[3]{x}}} \,\mathrm{d}x\]
Question 8(xi) \[\int \! {\frac{x+3}{\sqrt{7-6x-x^{2}}}} \,\mathrm{d}x\]
Question 8(xii) \[\int \! {\frac{3x-3}{\sqrt{x^{2}-2x+5}}} \,\mathrm{d}x\]