# 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

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

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