NSW HSC · Mathematics Advanced

Functions, Differential Calculus and Exponential Modelling

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

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Differentiate $y = (3 x^{2} - 5)^{4}$ with respect to $x$ .
Find the gradient of the normal to the curve $y = x^{2} - 4 x + 1$ at the point where $x = 3$ .
Find the equation of the horizontal asymptote of the function $f (x) = \frac{5 x - 3}{2 x + 7}$ .
A bacterial colony grows according to the model $N (t) = 500 e^{0.04 t}$ , where $t$ is measured in hours. What is the initial population of the colony?
What is the domain of the function $g (x) = ln (4 - x^{2})$ ?
Evaluate $lim_{x \to 0} \frac{s i n ( 5 x )}{x}$ .
Find $\frac{d}{d x} [x^{2} e^{x}]$ .
For what values of $x$ is the curve $y = x^{3} - 6 x^{2} + 9 x$ concave upwards?
Find the stationary points of the function $f (x) = x^{3} - 3 x$ .
Differentiate $y = tan (2 x)$ .
If the acceleration of a particle is given by $a (t) = 6 t - 4$ and its initial velocity is $v (0) = 5$ , find the velocity function $v (t)$ .
What is the exact value of $cos (\frac{2 π}{3})$ ?
Differentiate $y = e^{3 x^{2}}$ with respect to $x$ .
The function $f (x) = x ln x$ for $x > 0$ has a stationary point at:
A radioactive substance decays according to $N = N_{0} e^{- k t}$ , with $t$ in years. If its half-life is 10 years, the value of $k$ is:
Find the equation of the tangent to $y = ln (2 x - 1)$ at the point where $x = 1$ .
A square sheet of metal of side 12 cm has a square of side $x$ cm cut from each corner. The sides are then folded up to form an open box. What is the maximum volume of the box?
A population is modelled by $P (t) = 200 e^{0.05 t}$ , where $t$ is in years. After how many years, to one decimal place, will the population first reach 600?
How many points of inflection does the graph of $f (x) = x^{4} - 4 x^{3}$ have?
Differentiate $y = sin^{2} (3 x)$ .
Solve $2 sin^{2} x - sin x - 1 = 0$ for $0 \leq x \leq 2 π$ .

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Functions, Differential Calculus and Exponential Modelling

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