A ball is thrown upwards. The graph shows the height of the
ball in meters 𝑡 seconds after it was thrown. Work out an estimate for the
gradient of the graph when 𝑡 is equal to 0.8. Show all your working.
To find the gradient of the curve,
we first need to construct a tangent to the curve at the point given. On the 𝑥-axis of our graph, which
represents time, 10 little squares represent one second. That means one little square must
represent 0.1 seconds. 0.8 is therefore eight little
squares, as shown.
A tangent is a straight line which
touches a curve at one point only. Since we’re drawing this by eye,
this method will give us an approximate answer for the gradient at that point. Once we have constructed a tangent
to the curve that we are happy with, we can use the formula for the gradient of a
straight line to calculate the gradient of the graph at this point.
Gradient is change in 𝑦 divided by
change in 𝑥. You might also see this as 𝑦 two
minus 𝑦 one divided by 𝑥 two minus 𝑥 one. We will need then to find two
coordinates on our tangent. We can see that, on the 𝑦-axis,
which represents the height of the ball, five little squares represent one
meter. Dividing these both by five, we get
that one square is equal to 0.2 meters. The two coordinates that lie on our
tangent are therefore 0, 3.2 and 1.2, eight.
Remember, we said that the gradient
was the change in 𝑦 divided by the change in 𝑥. Our 𝑦-coordinates are 3.2 and
eight. So the change in 𝑦 is given by
eight minus 3.2. Our 𝑥-coordinates are zero and
1.2. So the change in 𝑥 is given by 1.2
We could have actually calculated
the gradient by subtracting these values the other way round. It doesn’t matter which direction
you choose to subtract as long as you are consistent. 3.2 minus eight over zero minus 1.2
will give us the same answer as our first calculation. Eight minus 3.2 is 4.8. And 1.2 minus zero is just 1.2.
We could type these into our
calculator. But if we were in a non-calculator
paper, we can multiply both the numerator and the denominator by 10 to give us 48
over 12. 48 divided by 12 is four. The gradient of the graph when 𝑡
is equal to 0.8 is four.
Fully describe what your answer to
part a represents. We found the gradient of the curve
when 𝑡 is equal to 0.8. Remember, we said that gradient is
change in 𝑦 divided by change in 𝑥. Looking at the context of our
question, we’ve worked out the gradient as the change in height, since height is the
𝑦-axis, over a time given, since the time is the 𝑥-axis.
We could also consider the change
in height as being a change in distance. So we’ve actually worked out
distance divided by time. We know that speed is equal to
distance over time. So we’ve actually worked out the
speed of the ball 0.8 seconds after it was thrown.
Explain why your answer to part a
is only an estimate. Remember, we drew our tangent by
eye. It’s not possible to draw the
tangent completely accurately. So this method will only give us an
approximate answer or an estimate for the gradient at that point.