Applications of Calculus

calculus year12 advanced

Applications of calculus | NSW Curriculum Website

• MAV-12-06 applies calculus to graph functions and model and solve problems involving optimisation, rates of change and motion in a line

📖 Prior Knowledge

Content	Prior knowledge	Used for
Area and Surface Area B	- area and surface area of various solids	- optimisation problems
Volume B	- volume of various solids	- optimisation problems
Equations C	- modelling a scenario using a quadratic equation	- optimisation problems
Introduction to Differentiation	- displacement, velocity, acceleration	- applications of differentiation and integration to motion
Differential Calculus	- derivative and its interpretation	- second derivative and its interpretation
Integral Calculus	- integral and its interpretation	- velocity and acceleration

Turning points, inflections and curve-sketching

• Define a function $f(x)$ to be differentiable at $x=a$ if \displaystyle \lim_\limits{h \to 0}\frac{f(a+h)-f(a)}{h} exists, that is if there is a non-vertical tangent to the curve at the point $P(a,f(a))$ and recognise that if $f(x)$ is differentiable at $x=a$ then it is continuous at $x=a$
• Identify any values of $x$ where a function is continuous, but not differentiable, given either the equation of the function or its graph
• Use repeated differentiation to find second derivatives of a function $y=f(x)$, denoting them by $f^{\prime \prime }(x)$ or $\frac{d^{2}y}{d{x}^2}$ or $y^{\prime \prime }$
• Analyse the stationary points of a function by testing values for $f^{\prime }(x)$, then classify the stationary points as local minima or local maxima where gradients change around the point, or horizontal points of inflection where gradients have the same sign on both sides of the point
• Interpret the second derivative $y^{\prime \prime }$ as the gradient function of the first derivative $y^{\prime }$, and deduce that if $y^{\prime \prime }>0$ the curve is concave up and if $y^{\prime \prime }<0$ the curve is concave down
• Define a point of inflection on a curve as a point where the concavity changes
• Analyse the value of $f^{\prime \prime }(x)$ either side of the roots of $f^{\prime \prime }(x)=0$, and use the resulting concavities to identify which zeroes of $f^{\prime \prime }(x)$ are points of inflection
• Use the second derivative to classify a stationary point as a local maximum, local minimum or a horizontal point of inflection
• Graph a function by determining local maxima and minima and points of inflection, horizontal and non-horizontal, considering any even or odd symmetry, the domain, any vertical asymptotes or other discontinuities, and where applicable, the behaviour of a function as $x\to \pm \infty$
• Graph $y=f^{\prime }(x)$ and $y=f^{\prime \prime }(x)$ for a function $y=f(x)$, given only a graph of $y=f(x)$

Optimisation

• Define a global maximum of a function $f(x)$ to be a point $P(a,f(a))$ on the graph where $f(x)\le f(a)$, for all $x$ in the domain, and define a global minimum similarly
• Examine whether any discontinuities or endpoints of the domain on which $f(x)$ is being considered are points of maxima or minima
• Model optimisation problems in a variety of contexts by defining variables, noting domain restrictions if necessary, and establishing functions to represent the relationship between variables
• Solve optimisation problems by using calculus to find local and global maxima and minima of differentiable functions, checking discontinuities of $f^{\prime }(x)$ and endpoints of the domain if applicable
• Formulate conclusions to optimisation problems by evaluating solutions given the constraints of the domain

Rates of change

• Use differentiation to find and interpret the first and second derivatives, $\frac{dQ}{dt}$ and $\frac{d^{2}Q}{d{t}^2}$, in practical problems where a quantity $Q$ is a function of time $t$
• Use integration to solve practical problems on the rate of change of a quantity $Q$, where $\frac{dQ}{dt}$ or $\frac{d^{2}Q}{d{t}^2}$ is given as a function of time $t$, together with sufficient initial conditions
• Describe and examine the graphs of practical situations where the rate of change of a quantity is proportional to the quantity
• Explain that if the rate of change of a positive quantity $Q$ over time $t$ is proportional to the size of $Q$, then this may be represented by $\frac{dQ}{dt}=kQ$ where $k$ is the constant of proportionality, and if $k>0$ the quantity $Q$ is increasing at a rate proportional to the value of $Q$ at time $t$, while if $k<0$ the quantity $Q$ is decreasing at a rate proportional to the value of $Q$ at time $t$
• Verify by substitution that the function $Q=A{e}^{kt}$ satisfies the relationship $\frac{dQ}{dt}=kQ$ with $A$ being the initial value of $Q$
• Recognise that the equation $\frac{dQ}{dt}=kQ$ where $Q>0$, and its solution $Q=A{e}^{kt}$ represent exponential growth when $k>0$ and exponential decay when $k<0$
• Graph the function $Q=A{e}^{kt}$, where $k>0$ and $k<0$ and $A>0$ and $A<0$, with and without graphing applications, and identify any asymptotes
• Model and solve growth and decay problems in various contexts using $\frac{dQ}{dt}=kQ$, $Q=A{e}^{kt}$ and the graph of $Q=A{e}^{kt}$ for $t\ge 0$, and justify conclusions in the context of the problem
• Determine the velocity and acceleration of a particle moving in a straight line given its displacement from a point as a function of time, and use the notation $\frac{d^{2}x}{d{t}^2}$ or $\ddot{x}$ to represent acceleration
• Solve problems relating to the motion of a particle moving in a straight line, using both differentiation and integration to connect the concepts of displacement, velocity and acceleration