Further Applications of Calculus

calculus year12 ext1

Further applications of calculus | NSW Curriculum Website

• ME1-12-05 applies calculus to solve problems involving polynomials, further rates of change, areas and volumes and differential equations

📖 Prior Knowledge

Content	Prior knowledge	Used for
Volume B	- volumes of cylinders, cones and spheres	- volumes of solids of revolution
Integral Calculus	- integration	- further areas and volumes of solids of revolution
Applications of Calculus	- first and second derivatives	- differential equations
Working with Functions	- solving simultaneous equations	- partial fractions

Multiplicity of zeroes of polynomial functions

• Use the product rule to prove that if $P(x)$ has a zero, $\alpha$, of multiplicity $m>1$, then $\alpha$ is a zero of $P^{\prime }(x)$ of multiplicity $m-1$, and use the result to determine the multiplicity of a discovered zero of $P(x)$ and solve related polynomial problems
• Prove that if $\alpha$ is a zero of multiplicity $m$, meeting the x-axis at $A(\alpha ,0)$, then if $m=1$, the curve crosses the x-axis at $A$ at an acute or obtuse angle; if $m>1$ is even, the curve is tangent to the x-axis at $A$ and does not cross it; if $m>1$ is odd, the curve has a horizontal inflection at $A$
• Graph a polynomial function in factored form, identifying its turning points and points of inflection if possible, and explore its behaviour as $x\to \infty$ and $x\to -\infty$, verifying the shape of the graph using graphing applications

Further rates of change

• Develop models in contexts where a rate of change of a function can be expressed as a rate of change of a composition of two functions, so that the chain rule can be applied
• Solve problems involving related rates of change using the chain rule, given the required formulas for problems relating to area, surface area or volume
• Describe and examine the graphs of practical situations where the rate of change of a quantity is proportional to the amount $Q-P$ by which the quantity $Q$ exceeds some fixed value $P$
• Explain that if the rate of change of a quantity $Q$ over time $t$ is proportional to the difference $Q-P$ at any instant, then this may be represented by the equation $\frac{dQ}{dt}=k(Q-P)$, where $k$ is a constant
• Verify by substitution that the function $Q=P+A{e}^{kt}$, where $A$ is a constant, satisfies the relationship $\frac{dQ}{dt}=k(Q-P)$, and that $Q=P$ in the case where $A=0$
• Graph the function $Q=P+A{e}^{kt}$, where $k>0$ and $k<0$ and $A>0$ and $A<0$, with and without graphing applications, and identify any asymptotes
• Use $\frac{dQ}{dt}=k(Q-P)$, $Q=P+A{e}^{kt}$ and the graph of $Q=P+A{e}^{kt}$ for $t\ge 0$, where $k>0$ or $k<0$ and $A>0$ or $A<0$, to model and solve problems where a limiting value of $Q$ exists, including Newton’s Law of Cooling and ecosystems with a natural carrying capacity, and justify conclusions in the context of the problem

Areas between curves and volumes of solids of revolution

• Calculate areas of regions between curves determined by functions in both real-life and abstract contexts
• Examine a solid of revolution whose boundary is formed by rotating an arc of a function about the x-axis or y-axis with and without graphing applications
• Calculate the volume of a solid of revolution formed by rotating a region in the plane about the x-axis or y-axis in both real-life and abstract contexts
• Calculate the volume of a solid of revolution formed by rotating the region between two curves about either the x-axis or y-axis in both real-life and abstract contexts

Differential equations

• Define a differential equation as an equation involving an unknown function and one or more of its derivatives
• Define and identify the order of a differential equation as the order of the highest derivative contained within the equation
• Recognise that a solution to a first order differential equations is a function, and that there may be infinitely many functions that are solutions to a given first order differential equation
• Recognise the solutions to differential equations in the context of slope fields, and that slope fields are useful in determining the behaviour of solutions when the differential equation cannot be easily solved
• Recognise that a unique solution of a differential equation can be determined when sufficient initial conditions are given, and refer to a problem involving a differential equation and initial conditions as an initial value problem (IVP)
• Graph solutions to first order differential equations given a slope field and identify the unique solution curve that satisfies a set of initial conditions
• Explore problems given a slope field representing a practical context and justify conclusions
• Form a slope field for a first order differential equation using graphing applications
• Recognise the features of a slope field corresponding to a first order differential equation and vice versa
• Solve first order differential equations of the form $\frac{dy}{dx}=f(x)$
• Solve first order differential equations of the form $\frac{dy}{dx}=g(y)$, where possible expressing the solution as a function with $y$ as the subject
• Recognise and solve the first order differential equations for exponential growth and decay: $\frac{dQ}{dt}=kQ$ and $\frac{dQ}{dt}=k(Q-P)$
• Solve first order differential equations of the form $\frac{dy}{dx}=f(x)g(y)$ using separation of variables, where possible expressing the solution as a function with $y$ as the subject
• Graph solutions of first order differential equations using graphing applications and examine the behaviour of solutions for different values of the constant of integration and initial conditions
• Solve differential equations of the form $\frac{dP}{dt}=kP\left(1-\frac{P}{C}\right)$ for some constants $k$ and $C$, given the appropriate decomposition into partial fractions, to obtain the logistic function
• Model and solve differential equations in practical scenarios including in chemistry, biology and economics