교육자를 위해: Gradients as rates of change(Mathematics, Ratio, proportion and rates of change)을(를) 위한 바로 쓸 수 있는 수업 슬라이드, 복습 노트 — 수업에 사용하거나, 학습자들이 실시간 게임으로 즐기는 인터랙티브 클래스 활동으로 진행하세요.
The gradient of a graph measures how fast y changes compared with x: \frac{\text{change in } y}{\text{change in } x}.
The units of the gradient are units of y per unit of x: for a distance–time graph, metres per second.
Straight-line graphs
A straight line has a constant rate of change. A line from (0, 2) to (4, 14) has gradient \frac{12}{4} = 3.
On a proportional currency-conversion graph, gradient is the example exchange rate in the plotted units. On a linear cost graph it is price per unit.
Curved graphs
A tangent has the same gradient as the curve at the chosen point. It can cross the curve at an inflection point or meet the curve elsewhere; its gradient gives the instantaneous rate.
The average rate of change between two points is the gradient of the chord joining them.
Average and instantaneous rates
Interpreting graphs
Compare steepness only on the same scales. The magnitude of gradient measures rate magnitude; its sign records increase or decrease. A horizontal line shows no change.
Area under a rate graph gives the total change: the area under a speed–time graph is distance.
Real contexts
A volume–time graph has gradient equal to volumetric flow rate. For constant incoming volume per second, depth rises faster where the container has a smaller horizontal cross-sectional area.
슬라이드
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1.What does the gradient of cumulative distance travelled against time represent?
Easy
Aspeed
Bdistance
Ctime
Dacceleration
2.A line passes through (0, 0) and (5, 20). What is the gradient?
Easy
A4
B5
C0.25
D100
3.A line passes through (2, 3) and (6, 11). What is the gradient?
Easy
A2
B4
C8
D0.5
4.An example proportional currency-conversion graph shows £10 = €12. What is the gradient in euros per pound?
Easy
A1.2
B0.83
C12
D120
5.The gradient of a speed–time graph is 3 m/s². What does this mean?
Easy
Ait accelerates at 3 m/s each second
Bit travels 3 m
Cit is at rest
Dspeed is constant
6.A horizontal line on a graph of temperature against time means:
Easy
Atemperature is rising
Btemperature is falling
Ctemperature is constant
Dtemperature is zero
7.What are the units of the gradient of a graph of distance (km) against time (hours)?
Easy
Akm/h
Bh/km
Ckm
Dhours
8.A graph of water volume (litres) against time (minutes) has gradient 5. What does it mean?
Easy
A5 litres per minute
B5 litres in total
C5 minutes per litre
D5 minutes
더 많은 문제
9.A line from (1, 10) to (5, 2) has gradient:
Medium
A-2
B2
C-0.5
D0.5
10.What does a negative gradient on a graph of fuel in a tank against time mean?
Medium
Athe fuel is decreasing
Bthe fuel is increasing
Cno fuel
Dconstant fuel
11.A tangent drawn to a curve at x = 2 passes through (0, 1) and (2, 5). What is the instantaneous rate of change at x = 2?
Medium
A2
B0.5
C4
D5
12.A curve passes through (1, 2) and (3, 10). What is the average rate of change between them?
Medium
A4
B8
C0.25
D6
13.A tank fills at 3 litres per minute. After 8 minutes it contains 30 litres. How much did it contain at the start?
Medium
A6 litres
B24 litres
C54 litres
D3 litres
14.On identical axis scales, which positive-sloping straight line shows the greatest positive rate of change?
Medium
Athe steepest line
Bthe flattest line
Cthe horizontal line
Dthe lowest line
15.A bottle is filled with water at a constant rate. Its width is narrow at the top. How does the graph of depth against time look near the top?
Medium
Asteeper
Bflatter
Chorizontal
Dunchanged
16.Population P (thousands) against time t (years) has a tangent at t = 4 through (2, 10) and (6, 18). What is the rate of growth at t = 4?
Medium
A2 thousand per year
B0.5 thousand per year
C8 thousand per year
D18 thousand per year
17.A speed–time graph is a straight line from (0, 4) to (5, 24). What is the acceleration?
Medium
A4 m/s²
B5 m/s²
C0.25 m/s²
D20 m/s²
18.A graph of cost against units of electricity has gradient 0.25 and intercept 12. What does the 12 represent?
Medium
Aa fixed charge
Bthe cost per unit
Cthe total cost
Dthe units used
19.On the same scales, a steeper graph of cumulative distance travelled against time shows:
Easy
Afaster the speed
Bslower the speed
Cgreater the time
Dsmaller the distance
20.What does the gradient of a chord joining two points of a curve represent?
Easy
Aaverage rate between two points
Binstantaneous rate
Carea
Dthe origin
21.A cumulative-distance–time curve has an increasing positive gradient. The object is:
Medium
Aspeeding up
Bslowing down
Cstationary
Dconstant speed
22.The gradient of a graph is the rate of change.
Easy
True or false?
23.A horizontal line has gradient zero.
Easy
True or false?
24.The gradient of a curve is the same at every point.
Easy
True or false?
25.A tangent shows the instantaneous rate of change.
Easy
True or false?
26.The units of the gradient are the same as the units of the y-axis.
Medium
True or false?
27.For straight-line journeys plotted as distance from the start against time, on fixed scales, which statements are true?
Medium
AThe magnitude of the gradient gives speed
Ba horizontal line shows no movement
CA steeper slope in magnitude shows greater speed
Darea is acceleration
Ea negative gradient shows moving towards the start
28.Which can be found from a tangent to a curve?
Medium
Athe instantaneous rate of change
Bthe gradient at a point
Cthe average rate between two points
Dthe area under the curve
Ethe equation of the tangent line
29.A graph of money (£) against hours worked is a straight line through (0, 0) and (4, 36). What is the hourly rate?
Easy
A£9
B£4
C£144
D£0.11
30.A graph of temperature against time has gradient -2 °C per minute. What is happening?
Easy
Athe temperature falls by 2 °C each minute
Bit rises by 2 °C each minute
Cit stays constant
Dit is at 2 °C
31.Two positive-sloping straight lines use the same axis scales. Line A is steeper. Which has the greater positive rate of change?
Easy
AA
BB
Cthey are equal
Dcannot tell
32.A curve passes through (2, 3) and (6, 11). Find the average rate of change between these points, and explain how you would find the rate of change at x = 4.
Medium
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