Estimating Gradients & Areas under Graphs Flashcards

All 12 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$^2$.

What are the steps to estimate the area under a curve using strips of equal width?

e.g. the area under a speed–time curve from $t = 0$ to $t = 6$ using 3 strips

1. Split the region into strips of equal width: width $= 2$
2. Treat each strip as a trapezium, area $= \frac{1}{2} \times \text{width} \times (y_1 + y_2)$: e.g. $\frac{1}{2} \times 2 \times (25 + 16) = 41$
3. Add the strip areas: $41 + 25 + 13 = 79$

What is the formula for the area of one trapezium strip used to estimate the area under a curve?

e.g. a strip of width $1$ with heights $4$ and $6$

$\frac{1}{2} \times \text{width} \times (y_1 + y_2)$, where $y_1$ and $y_2$ are the two heights.

e.g. $\frac{1}{2} \times 1 \times (4 + 6) = 5$

What does the area under a speed–time (velocity–time) graph represent?

The distance travelled.

e.g. speed in m/s and time in s give a distance in metres.

True or false? On a distance–time graph, the area under the graph represents the speed.

False. Speed is given by the gradient; the area under a distance–time graph has no useful meaning.

On a graph of the rate of flow of water (litres per second) against time (seconds), what does the area under the graph represent?

The total volume of water that has flowed, in litres.

An estimate for the area under a curve is worked out using trapezium strips, and the straight top of each strip lies above the curve. Is the estimate an underestimate or an overestimate, and why?

An overestimate. The area between the trapezia and the curve is also included.

True or false? If the straight top of each trapezium strip lies below the curve, the estimate for the area under the curve is an overestimate.

False. It is an underestimate, because the area between the trapezia and the curve is not included.