The Gradient of a Curve as a Limit

Mathematics · Calculus & Analysis · ages 16-17

Name ______________________ Date ____________

Setting h = 0 puts the second point exactly on top of the first one.
Circle one: True False
The gradient of a chord is equal to the gradient of the curve at the point where the chord starts.
Circle one: True False
A set of chord gradients runs 7, then 6.1, then 6.01, then 6.001. What number are they closing in on?
- 7
- 6.001
- 6
- 0
On y = x², what is the gradient of the chord joining (1, 1) and (4, 16)?
Answer: ______________
On y = x², what is the gradient of the chord joining (3, 9) and (3.5, 12.25)?
Answer: ______________
On y = x² at x = 2, a chord to the left with h = -0.1 has gradient 3.9, and a chord to the right with h = 0.1 has gradient 4.1. What does that pair tell you?
- the curve has two different gradients at x = 2
- the gradient at x = 2 is caught between them, at 4
- the left-hand chord must have been worked out wrongly
- the curve is a straight line between x = 1.9 and x = 2.1
On y = x², the gradient of the curve at x = 3 is bigger than the gradient at x = 1.
Circle one: True False
On y = x², chords at x = 6 give gradients of 12.1 when h = 0.1 and 12.01 when h = 0.01. What is the gradient of the curve at x = 6?
Answer: ______________
On y = x², what is the gradient of the chord from x = 4 to x = 4.01?
- 8.01
- 8.1
- 8
- 16.01
Kai says that because the chord gradients get closer and closer to 4 but never equal 4, the gradient at the point cannot really be 4. Where is the slip?
- the chord gradients do reach 4, at h = 0
- the gradient at the point is the average of the chord gradients instead
- a gradient that is never reached does not exist at all
- the limit is the value they close in on, and that value is the gradient

Answer key

For grown-ups. Fold this page away before handing over the rest.

The Gradient of a Curve as a Limit W1-mt_LAlhcwVjln-s1

True · With h = 0 there is no gap in x left at all, so the two ends of the chord land in the same place and there is nothing left to join.
False · At x = 2 the chord with h = 1 gives 5 while the gradient at the point is 4. Shrinking h narrows the gap but never closes it.
6 · Each value is nearer 6 than the last, and the gap keeps shrinking by a factor of ten. The limit is 6.
5 · Divide the change in y by the change in x: (16 - 1) / (4 - 1) = 15 / 3 = 5.
6.5 · The rise is 12.25 - 9 = 3.25 and the run is 0.5, so the gradient is 3.25 / 0.5 = 6.5.
the gradient at x = 2 is caught between them, at 4 · The left-hand chords climb towards the answer and the right-hand ones fall towards it. Both sides agree on 4, which is strong evidence that 4 is the gradient.
True · Chords on either side of x = 3 come out steeper than chords on either side of x = 1, because this curve climbs faster the further right you go.
12 · The gradients are closing in on 12, each one ten times nearer than the last. The limit is 12.
8.01 · At x = 4.01, y = 16.0801. The rise is 0.0801 and the run is 0.01, so the gradient is 0.0801 / 0.01 = 8.01.
the limit is the value they close in on, and that value is the gradient · No chord is asked to equal 4. The gradient at a point is defined as the value the chords approach, and approaching 4 from both sides is what pins it down.