ESAT (Engineering and Science Admissions Test) · Mathematics 2

Differentiation

11 practice questions, each with the answer and an explanation.

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Random questions from this topic, one at a time. You see the answer and the explanation straight after each one.

All 11 questions

What are the coordinates of the local maximum of $y = x^{3} - 6 x^{2} + 9 x + 1$ ?
What is the equation of the normal to $y = x$ at the point where $x = 4$ ?
What is the gradient of $y = (2 x + 1)^{5}$ at $x = 0$ ?
Differentiate $y = ln (x^{2} + 1)$ .
Differentiate $y = sin (x^{2})$ .
Differentiate $y = x^{2} e^{3 x}$ with respect to $x$ .
What is the equation of the tangent to $y = x^{3} - 3 x$ at the point where $x = 2$ ?
A square sheet of side $12$ cm has squares of side $x$ cut from its corners and the sides folded up to make an open box. What is the greatest possible volume?
For which values of $x$ is $f (x) = 2 x^{3} - 9 x^{2} + 12 x$ an increasing function?
A spherical balloon's radius increases at $0.5 cm/s$ . How fast is its volume increasing when the radius is $4 cm$ ? ( $V = \frac{4}{3} π r^{3}$ )
The curve $y = x^{4} - 4 x^{3}$ curves upwards where $\frac{d ^{2} y}{d x ^{2}} > 0$ . For which values of $x$ does this hold?

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