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NSW Y12 Maths - Advanced Differential Calculus Primitive Functions

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Primitive Functions Theory

Finding a primitive function is the reverse of differentiating a function.\\ when finding a primitive function (integrating) from a given function, it is important to include the constant '\(c\)' and in some cases determine the value of '\(c\)'.\\ \textbf{Example}\\ %17220 The gradient of the tangent to a curve is given by \(\dfrac{{dy}}{{dx}} = 4 - 6x\) and the curve passes through the point \((-1,1)\). The equation of the curve is?\\ \textbf{Solution}\\ $\begin{aligned} \frac{d y}{d x} &=4-6 x \\ y &=4 x-6 \frac{x^{2}}{2}+c \\ y &=4 x-3 x^{2}+c \\ \text { At }\ (-1,1)\ \ 1&=-4-3+c \\ c&=8\\ \therefore\ y&=-3 x^{2}+4 x+8 \end{aligned}$\\

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