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MTH1020 - Analysis of change - S2 2025

MTH1020 - Analysis of change - S2 2025

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Consider the functions f(x) = 2x^3 and g(x) = 3x^5.

Choose the correct expression for the derivative of $\Big(\dfrac{f}{g}\Big)(x)$ \Big(\dfrac{f}{g}\Big)(x), that is, for $\Big(\dfrac{f}{g}\Big)'(x)$ \Big(\dfrac{f}{g}\Big)'(x).

-4x^2

❌

$-\dfrac{4}{3x^3}$ -\dfrac{4}{3x^3}

✅

$\dfrac{2}{3x^2}$ \dfrac{2}{3x^2}

❌

$\dfrac{2}{3x^3}$ \dfrac{2}{3x^3}

❌

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Consider the functions f(x) = 2x^2 and g(x) = x^7.

Choose the correct expression for the derivative of $\big(f\,g\big)(x)$ \big(f\,g\big)(x), that is, for $\big(f\,g\big)'(x)$ \big(f\,g\big)'(x).

2x^9

14x^7

56x^8

18x^8

Which of the following statements regarding differentiability and continuity of functions are true? Choose the correct statement/s.

Differentiability implies continuity.

✅

If a function is continuous for all x, it will also be differentiable for all x.

❌

If a function is not continuous, it will also be not differentiable.

✅

Continuity implies differentiability.

❌

If a function is differentiable for all x, it will also be continuous for all x.

✅

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Which of the following functions are continuous for all x but not differentiable for all x?

$f(x) = \sqrt[3]{x}$ f(x) = \sqrt[3]{x}

$k(x)=\left\{\begin{array}{ll}x+1&\text{ for }x < 1\\3-x&\text{ for }x \geq 1\end{array}\right.$ k(x)=\left\{\begin{array}{ll}x+1&\text{ for }x < 1\\3-x&\text{ for }x \geq 1\end{array}\right.

$h(x) = \dfrac{1}{x^2+4}$ h(x) = \dfrac{1}{x^2+4}

$g(x) = x\sin(x)$ g(x) = x\sin(x)

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Which of the following functions are differentiable for all x?

$f(x) = 2x\cos(x+1)$ f(x) = 2x\cos(x+1)

✅

h(x) = |x^2-2x|

❌

$g(x) = e^{\sin(x)}$ g(x) = e^{\sin(x)}

✅

$t(x)=\left\{\begin{array}{ll}1-x&\text{ for }x < 0\\1-x^2&\text{ for }x \geq 0\end{array}\right.$ t(x)=\left\{\begin{array}{ll}1-x&\text{ for }x < 0\\1-x^2&\text{ for }x \geq 0\end{array}\right.