Section 33.1

Calculus

Derivatives, integrals, related rates, optimisation and volumes.

Toolkit

These problems combine standard calculus tools with tricks for handling parameters cleverly.

Logarithmic differentiation: for $y=f(x)^{g(x)}$ , use $\ln y = g(x)\ln f(x)$ , then $y'=y\left(g'\ln f + g f'/f\right)$ .
Related rates: differentiate a constraint $F(x(t),y(t))=0$ implicitly in $t$ , then plug in the snapshot values of $x,y,\dot x$ (not before differentiating).
Comparing exponentials: $a^b$ vs $b^a$ reduces to comparing $\ln a/a$ vs $\ln b/b$ ; $\ln x/x$ increases on $(0,e)$ and decreases on $(e,\infty)$ .
Optimisation: set derivative to zero, check endpoints/second derivative; for a cubic in one variable, factor or use the discriminant.
Volumes by slicing: $V=\int A(x)\,dx$ where $A(x)$ is the cross-sectional area; symmetric solids often have simple (square, circular) cross-sections.
Operator method for linear ODEs: write $f-f'=(1-D)f$ ; if the right side is a degree- $n$ polynomial, invert as $f=(1+D+D^2+\cdots)(\text{poly})$ , a finite sum, then fix constants with initial data.
Limits of integrals with a parameter: substitute to expose the parameter’s role (e.g. $y=x^n$ ), then interchange limit and integral when justified (dominated convergence or direct bounding).