Calculus Techniques Interactive Diagrams

Tangent line given function and \(x\)-coordinate

Note that the results shown are based on numeric approximations and should be taken as illustrative only.

Find the equation of the tangent to the curve \(f(x)=\) where \(x=\)

1. Evaluate \(f(x)\) to determine coordinates of the point the tangent must pass through
2. Determine \(f'(x)\)
3. Evaluate \(f'(x)\) at the given value of \(x\) to determine gradient of tangent line
4. Use gradient and point give the equation of the line

Tangent line from parametric equations

Find the equation of the tangent to the curve given by parametric equations \(x(t)=\) and \(y(t)=\) where \(t=\)

(Graph over the domain \(\leq t\leq\))

1. Evaluate \(x(t)\) and \(y(t)\) to determine coordinates of the point the tangent must pass through \(A:\left(x(t),y(t)\right)\)
2. Determine \(\frac{dy}{dx}=\frac{dy}{dt}\frac{dt}{dx}\)
3. Evaluate \(\frac{dy}{dx}\) at the given value of \(t\) to determine gradient of tangent line
4. Use gradient and point to give the equation of the line

Area under a curve

Find the area between the curve \(f(x)=\) and the \(x\)-axis over the interval to

Area between curves

Find the area between the curve \(f(x)=\) and \(g(x)=\) over the interval to