Hello! Last lesson, you built exponential models from a starting value and a constant percentage change. You also saw that a graph can grow steeper even when its percentage change stays constant. Today the question is more local: how quickly is a quantity changing at one particular moment? You will estimate that rate from a graph by drawing a tangent and finding its gradient. This is useful both in Math AA and when interpreting graphs of experimental data.
A rate at one point
If a graph shows distance against time, the slope between two plotted points gives an average speed over an interval. On a curved graph, that average depends on which interval you choose. To estimate the speed at one instant, you need the slope of the curve at that point.
A tangent is a straight line aligned with the curve at the chosen point. Its gradient estimates the curve’s instantaneous rate of change there. Watch how the tangent changes as its point moves along the graph.
Finding the gradient of a curve by drawing a tangent
Watch “Finding the gradient of a curve by drawing a tangent,” by Mark Willis, to see why one curved graph can have different gradients at different points.
In the moving tangent, focus on how its slope changes from negative to zero to positive as the point travels along the quadratic curve. The tangent belongs to a specified point, not to the curve as a whole.
This gives a useful link to the earlier lesson on quadratic maxima and minima: at a smooth turning point, the tangent is horizontal, so its instantaneous rate of change is zero. On an increasing part of a curve the tangent slopes upwards; on a decreasing part it slopes downwards.
The distinction is worth keeping clear. A line joining two points on the curve is a secant and measures an average rate over an interval. A tangent is positioned at one target point and estimates the rate there. We will use ordinary straight-line gradient calculations once the tangent has been drawn.
Draw the tangent, then measure its gradient
Suppose you are asked for the instantaneous rate at . First locate the point on the curve whose horizontal coordinate is 3. Place a ruler through that point and rotate it until its edge follows the curve’s direction near the point. Then draw a reasonably long straight line.
The common description that a tangent “touches without cutting” is a helpful guide for many graphs, but the important test is local alignment at the chosen point. You are not trying to join two convenient points on the curve. Nor must the drawn line remain on one side of the curve everywhere else.
Finding the gradient of a curve by drawing a tangent
Continue the same video for a ruler-based demonstration and a check on what a horizontal tangent means.
Watch the ruler method, from locating the point at x=3 through calculating the gradient. Notice the warning to read axis values, rather than treating distances on the page as numerical values. Then watch the zero gradient as the presenter identifies a horizontal tangent at a turning point.
Once you have extended the tangent, choose two easy-to-read points on the tangent line. They can be well away from the target point, and they need not lie on the curve. If their coordinates are and , calculate
Choosing points far apart on the drawn tangent usually makes small reading errors less significant. A large, clear “rise over run” triangle is more reliable than a tiny one crowded around the point of contact.
Read the triangle on the line
The diagram below shows a blue curve and a red tangent at . The dashed triangle is drawn using points on the red line.
Here the two convenient points on the tangent are and . Its gradient is
Thus the curve’s instantaneous rate of change at is approximately 2 units of per unit of . It is an estimate because the tangent was positioned and read from a drawing.
Notice a subtle but essential detail: the point is on the red tangent, not necessarily on the blue curve. Reading two points from the blue curve instead would calculate a secant gradient—an average rate—not the gradient of the tangent you drew.
Always check the axes before calculating. If one grid square represents 2 minutes horizontally and 10 degrees vertically, a triangle spanning three squares each way has a run of 6 minutes and a rise of 30 degrees. Its gradient is 5 degrees per minute, not degree per minute.
From graphical gradient to a meaningful rate
A reaction graph makes the units and interpretation concrete. If the vertical axis records the volume of gas produced in cubic centimetres and the horizontal axis records time in seconds, then a tangent gradient has units of cubic centimetres per second.
Drawing a tangent to find instantaneous rate of reaction
Watch “Drawing a tangent to find instantaneous rate of reaction,” by Physics with Simon Poliakoff, for an applied example that separates the rate at one time from the average rate so far.
Watch the changing rate to see why the curve’s steepness matters. In placing the tangent, follow the choice of the five-second point. Then watch the large triangle for the coordinate differences and units. Finally, average versus instant explains why dividing the five-second reading by five does not answer the question about the rate at five seconds.
In that demonstration, the drawn tangent rises by about over a run of . Its estimated gradient is therefore
This says that at the chosen moment, gas volume is increasing at approximately per second. It does not say that exactly was produced in every one-second interval of the reaction. The curve changes steepness, so its rate changes with time.
Rates can also be negative. For example, if a cooling graph has temperature on the vertical axis and time on the horizontal axis, a downward-sloping tangent has a negative gradient in degrees per minute. The minus sign tells you the temperature is falling; the size of the gradient tells you how quickly.
When reporting a graphical estimate, show the two tangent-line coordinates you used, the vertical and horizontal differences, and the resulting value with units. A slightly different careful tangent may give a slightly different estimate. What matters is that the line follows the curve near the requested point and that the arithmetic matches your drawn line.
Conclusion
To estimate an instantaneous rate, locate the specified point, draw a line aligned with the curve there, and calculate that tangent’s gradient using two well-separated points on the line. Keep an average rate from two curve points distinct from a rate at one point, and interpret the sign and units using the graph’s axes.
Next, you will examine another feature of experimental data: how spread out repeated measurements are, using standard deviation.