Topic 5: Calculus

Topic 5 is the heart of analysis in IB Mathematics: Analysis and Approaches. It begins with the limit, the precise idea behind the gradient of a curve at a single point, and develops this into the derivative $\\frac{dy}{dx}$. From there you learn rules that let you differentiate almost any expression you meet, and you apply differentiation to real problems: finding the equation of a tangent line, locating maxima and minima, and modelling motion. The second half reverses the process. Integration recovers a function from its rate of change and, through the definite integral, measures the area under a curve and the net change of a quantity over time. SL covers the core ideas and rules; HL adds the chain, product and quotient rules (in their full form), more standard derivatives, and integration by substitution. Master the algebra of indices and the standard function library first, because most calculus errors are really algebra errors in disguise.

Limits and the derivative

Calculus starts with a limit, written $\\lim_{x\\to a} f(x)$, which describes the value a function approaches as $x$ gets close to $a$ (not necessarily its value at $a$). The gradient of a curve at a point is found by taking a chord between two nearby points and letting them slide together. If the second point is a horizontal distance $h$ away, the chord gradient is $\\frac{f(x+h)-f(x)}{h}$, and the derivative is the limit of this as $h\\to 0$. This is differentiation from first principles: $f'(x)=\\lim_{h\\to 0}\\frac{f(x+h)-f(x)}{h}$. The result is a new function, the derivative, which gives the gradient of the original curve at any value of $x$. Two notations mean the same thing: Lagrange notation $f'(x)$ and Leibniz notation $\\frac{dy}{dx}$. A positive derivative means the function is increasing, a negative derivative means it is decreasing, and a zero derivative means the gradient is momentarily flat. You should be able to apply first principles to a simple polynomial such as $f(x)=x^{2}$, where the working gives $f'(x)=2x$.

Differentiation rules

In practice you differentiate using rules rather than first principles. The power rule is the workhorse: if $y=x^{n}$ then $\\frac{dy}{dx}=nx^{n-1}$, for any rational $n$. A constant multiple is carried through, $\\frac{d}{dx}(kx^{n})=knx^{n-1}$, and sums are differentiated term by term. The derivative of a constant is $0$. To use the power rule you often rewrite expressions first: a root becomes a fractional index, $\\sqrt{x}=x^{1/2}$, and a reciprocal becomes a negative index, $\\frac{1}{x^{2}}=x^{-2}$. For example $\\frac{d}{dx}\\left(3x^{4}-\\frac{2}{x}\\right)=12x^{3}+\\frac{2}{x^{2}}$. HL students also need three combination rules. The chain rule differentiates a composite function: if $y=g(u)$ and $u=f(x)$ then $\\frac{dy}{dx}=\\frac{dy}{du}\\times\\frac{du}{dx}$. The product rule handles a product: if $y=uv$ then $\\frac{dy}{dx}=u\\frac{dv}{dx}+v\\frac{du}{dx}$. The quotient rule handles a fraction: if $y=\\frac{u}{v}$ then $\\frac{dy}{dx}=\\frac{v\\frac{du}{dx}-u\\frac{dv}{dx}}{v^{2}}$. Choosing the right rule, and combining them when needed, is a core HL skill.

Derivatives of trig, exponential and log functions

Beyond powers you must know a small library of standard derivatives. For trigonometric functions (with $x$ in radians) $\\frac{d}{dx}(\\sin x)=\\cos x$ and $\\frac{d}{dx}(\\cos x)=-\\sin x$; HL adds $\\frac{d}{dx}(\\tan x)=\\sec^{2}x$. The exponential function is special because it is its own derivative: $\\frac{d}{dx}(e^{x})=e^{x}$. The natural logarithm gives $\\frac{d}{dx}(\\ln x)=\\frac{1}{x}$. At HL these combine with the chain rule to give general forms such as $\\frac{d}{dx}(e^{f(x)})=f'(x)e^{f(x)}$ and $\\frac{d}{dx}(\\ln f(x))=\\frac{f'(x)}{f(x)}$, and for example $\\frac{d}{dx}(\\sin(3x))=3\\cos(3x)$. Many exam questions mix these standard derivatives with the product or quotient rule, for instance differentiating $x^{2}e^{x}$ gives $2xe^{x}+x^{2}e^{x}$. Always check whether the angle is in radians, because the neat trig derivatives only hold in radian measure.

Tangents and normals

The derivative gives the gradient of a curve at a point, which lets you write the equation of the tangent line there. The tangent at the point $(a, f(a))$ has gradient $m=f'(a)$, and its equation comes from the point-gradient form $y-f(a)=f'(a)(x-a)$. The normal is the line perpendicular to the tangent at the same point. Perpendicular gradients multiply to $-1$, so the normal has gradient $-\\frac{1}{f'(a)}$ (provided $f'(a)\\neq 0$), and its equation is $y-f(a)=-\\frac{1}{f'(a)}(x-a)$. A reliable method is: differentiate to get $f'(x)$, substitute the $x$-coordinate to find the gradient, find the $y$-coordinate from the original function, then build the line. On the calculator paper you can often read tangent gradients or equations directly, but you should still be able to do the full algebra by hand for the non-calculator paper.

Stationary points, the second derivative and optimization

A stationary point is where the gradient is zero, found by solving $f'(x)=0$. Stationary points are classified as local maxima, local minima, or points of inflexion. One way to classify is the sign of $f'(x)$ either side of the point. A cleaner way uses the second derivative $f''(x)$, obtained by differentiating $f'(x)$ again. If $f''(x)\\lt 0$ at a stationary point the curve is concave down and the point is a local maximum; if $f''(x)\\gt 0$ the curve is concave up and it is a local minimum; if $f''(x)=0$ the test is inconclusive and you fall back on the sign of the first derivative. Points of inflexion, where concavity changes, satisfy $f''(x)=0$ with a genuine sign change. Optimization applies all of this to real problems: write the quantity to be optimised as a function of one variable (using a constraint to eliminate any others), differentiate, set the derivative to zero, solve, and confirm whether you have a maximum or minimum. Typical contexts are maximum volume, minimum surface area, or least cost. Always answer the question that was asked, for example stating the maximum volume itself, not just the value of $x$ that produces it.

Integration and the indefinite integral

Integration reverses differentiation, so it is also called finding the antiderivative. The reverse power rule is $\\int x^{n}\\,dx=\\frac{x^{n+1}}{n+1}+c$, valid for $n\\neq -1$, where $c$ is the constant of integration that must always be included for an indefinite integral. For example $\\int x^{2}\\,dx=\\frac{x^{3}}{3}+c$. Constant multiples and sums behave just as in differentiation. You also reverse the standard derivatives: $\\int \\cos x\\,dx=\\sin x+c$, $\\int \\sin x\\,dx=-\\cos x+c$, $\\int e^{x}\\,dx=e^{x}+c$, and the case that the power rule misses, $\\int \\frac{1}{x}\\,dx=\\ln|x|+c$. HL extends integration with the reverse chain rule, integration by substitution. Here you replace part of the integrand with a new variable $u$, replace $dx$ using $\\frac{du}{dx}$, integrate in terms of $u$, and substitute back. For instance $\\int 2x(x^{2}+1)^{3}\\,dx$ is solved with $u=x^{2}+1$, giving $\\frac{(x^{2}+1)^{4}}{4}+c$. If a boundary condition such as a known point is given, substitute it after integrating to find $c$.

Definite integrals, area and kinematics

A definite integral has limits and produces a number rather than a function: $\\int_{a}^{b} f(x)\\,dx=F(b)-F(a)$, where $F$ is any antiderivative of $f$. This is the fundamental theorem of calculus, linking the two halves of the topic. Geometrically, $\\int_{a}^{b} f(x)\\,dx$ gives the signed area between the curve and the $x$-axis from $x=a$ to $x=b$; area below the axis counts as negative. To find an actual physical area that dips below the axis, integrate the regions separately or use $\\int_{a}^{b} |f(x)|\\,dx$. The area between two curves is $\\int_{a}^{b} (\\text{upper}-\\text{lower})\\,dx$. Kinematics is a major application. Displacement $s$, velocity $v$ and acceleration $a$ are linked by differentiation in one direction and integration in the other: $v=\\frac{ds}{dt}$ and $a=\\frac{dv}{dt}$, while $s=\\int v\\,dt$ and $v=\\int a\\,dt$. The distance travelled between times $t_1$ and $t_2$ is $\\int_{t_1}^{t_2} |v|\\,dt$, whereas the displacement (net change in position) is $\\int_{t_1}^{t_2} v\\,dt$; the two differ whenever the object changes direction. A particle is momentarily at rest when $v=0$ and its speed is greatest when acceleration $a=0$.