The average rate of change over \([x_0,x_0+h]\) is the slope of the secant line through two points. As \(h\to 0\), the secant pivots toward the tangent line at \(x_0\). The slope of this limiting tangent is the derivative \(f'(x_0)\).

Click the graph to place \(x_0\), then press Animate h→0 to watch the secant automatically converge to the tangent.

The first derivative \(f'(x)\) is positive where \(f\) rises, negative where it falls, and zero at critical points. The second derivative \(f''(x)\) captures concavity — where concavity switches we have an inflection point.

Toggle the three curves and click the graph to move the trace point.

The Closed Interval Method: global extrema on \([a,b]\) occur only at endpoints or at critical points inside \((a,b)\) where \(f'=0\). Evaluate \(f\) at all candidates and compare. Drag the interval bounds to see how the extrema shift.

A Riemann sum approximates area by partitioning \([a,b]\) into \(n\) subintervals of width \(\Delta x=(b-a)/n\) and placing a rectangle of height \(f(x_i^*)\) on each. Four methods — Left, Right, Midpoint, Trapezoid — all converge to \(\int_a^b f\,dx\) as \(n\to\infty\), with midpoint and trapezoid converging ~4× faster.

The definite integral \(\int_a^b f(x)\,dx\) measures signed area. Regions above contribute positively; below negatively. The integral returns the net result — they can cancel. If \(f(t)\) is velocity, the integral gives displacement (net position change).

Part 1: \(F(x)=\int_a^x f(t)\,dt\Rightarrow F'(x)=f(x)\). The running total's slope equals \(f\). Part 2: \(\int_a^b f\,dx=F(b)-F(a)\) — just evaluate any antiderivative at the bounds and subtract. Click or drag \(x\) to watch \(F(x)\) accumulate signed area.

When substituting \(x=a\) into \(f(x)/g(x)\) gives \(0/0\) or \(\infty/\infty\), the limit is indeterminate. L'Hôpital's Rule says we may replace the ratio with \(f'(x)/g'(x)\) and evaluate again. Both curves approach the same limit \(L\) as \(x\to a\).

The dashed curve is \(f/g\) (the original ratio). The solid curve is \(f'/g'\) (after applying the rule). Both converge to the same \(L\).

When a curve can't be written as \(y=f(x)\), we differentiate both sides with respect to \(x\), treating \(y\) as an implicit function and applying the chain rule to every \(y\)-term. The result: \(dy/dx=-F_x/F_y\).

Drag the angle slider to trace a point around the curve and read the tangent slope. The tangent is undefined at vertical tangent points where \(F_y=0\).

Related rates uses implicit differentiation with respect to time: differentiate both sides of a formula, substitute known rates, and solve for the unknown rate. Drag the slider to animate the geometry and read the computed rate.

A Taylor series centered at \(a\): \(T_n(x)=\sum_{k=0}^n\frac{f^{(k)}(a)}{k!}(x-a)^k\). Each additional term extends the radius of accurate approximation. Increase \(n\) to watch the solid polynomial track the dashed true function farther from center.

Rotating \(y=f(x)\) around the \(x\)-axis sweeps out a solid. The disk method slices it into circular disks of radius \(r=f(x)\) and area \(\pi r^2\). Integrating gives \(V=\pi\int_a^b[f(x)]^2\,dx\). Click or drag the slice slider to move the cross-sectional disk.

An infinite series \(\sum a_n\) converges if its partial sums \(S_n\) approach a finite limit. Geometric series \(\sum r^n\) converge when \(|r|<1\) with sum \(r/(1-r)\). The \(p\)-series \(\sum 1/n^p\) converges when \(p>1\) — the harmonic series (\(p=1\)) diverges astonishingly slowly.

To find the area between \(y=f(x)\) and \(y=g(x)\), integrate the absolute difference: \(A=\int_a^b|f(x)-g(x)|\,dx\). Find natural bounds by solving \(f(x)=g(x)\). When curves cross, split the integral and sum absolute sub-areas.

A slope field draws a short segment of slope \(f(x,y)\) at each point, sketching tangent directions any solution must follow. Given an initial condition \(y(x_0)=y_0\), a unique solution curve threads through that point. Click the graph to move the initial condition.

Integration by Parts is the antiderivative counterpart of the product rule: \(\int u\,dv=uv-\int v\,du\). The LIATE rule guides the choice of \(u\) — Logarithms, Inverse trig, Algebraic, Trig, Exponential. Blue area = \(\int u\,dv\); orange = \(\int v\,du\). Together they tile the rectangle \([uv]_a^b\).

We approximate the arc by \(n\) chord segments. Each chord of width \(\Delta x\) has length \(\sqrt{\Delta x^2+\Delta y^2}=\sqrt{1+(\Delta y/\Delta x)^2}\,\Delta x\). As \(\Delta x\to 0\) this sum becomes \(\int_a^b\sqrt{1+[f'(x)]^2}\,dx\). Drag \(n\) to watch the chord approximation converge.