Finding Gradients of Tangents Flashcards

All 5 cards in this deck

What is a tangent to a curve?

A straight line that touches the curve at one point and has the same gradient as the curve at that point.

What are the steps to estimate the gradient of a curve at a point?

e.g. estimate the gradient of a curve at $t = 3$

1. Draw a tangent to the curve at that point: a line touching the curve at $t = 3$
2. Read off two points on the tangent to get the change in $y$ and change in $x$: e.g. $\Delta y = 50$, $\Delta x = 2$
3. Gradient $= \dfrac{\Delta y}{\Delta x} = \dfrac{50}{2} = 25$

What does the gradient of a chord joining two points on a curve represent?

e.g. the chord from $t = 2$ to $t = 6$ on a distance–time curve

The average rate of change between those two points.

e.g. the average speed between $t = 2$ and $t = 6$.

On a distance–time graph, what does the gradient at a point represent, and in what units?

The speed (instantaneous speed) at that time, in distance units per time unit.

e.g. metres and seconds give m/s.

On a speed–time (velocity–time) graph, what does the gradient at a point represent, and in what units?

The acceleration (rate of change of speed) at that time.

e.g. speed in m/s and time in s give m/s$^2$.