This lesson aims to introduce calculus and its applications, whilst also introducing the concept of a gradient. By mastering the content in this session, you will:
Understand that calculus is the study of how a change in the input to a function effects its output
Be able to identify a few applications where calculus can be useful
Distinguish between continuous and discontinuous functions, identifying key elements which distinguish each type of function – Sketch graphs of functions that are continuous and compare them with graphs of functions that have discontinuities – Describe continuity informally, and identify continuous functions from their graphs
Examine and use the relationship between the angle of inclination of a line or tangent, 𝜃, with the positive 𝑥-axis, and the gradient, 𝑚, of that line or tangent, and establish that
C1.2 Difference Quotients
Interpret the meaning of the gradient of a function in a variety of contexts, for example on distance–time or velocity–time graphs
In simple terms, differentiation is the study of rates. When studying differentiation we are interested in the rate which a functions output changes with respect to its input.
Understanding rates is extremely useful when analysing real world functions. Some examples are shown below:
Examples Of Real World Input / Output Pairs
Profits vs Time Graph: These graphs could be used to represent the profits of a company over time. The rate would represent how effective that month was for money generation.
Distance vs Time Graph: These graphs could be used to represent a cars journey over time. The rate is effectively the speed at any point in the journey.
Height vs Pressure Graph: These graphs could be used to represent the pressure on a submarines walls as it submerges. The rate would represent the how much the pressure would increase at a particular depth.
Rates and Gradients
If a function is drawn on a cartesian plane, the gradient at any point represents the rate at which the function is changing.
A gradient of zero corresponds to no rate of change
A positive gradient indicates a increasing rate of change
A negative gradient indicates a decreasing rate of change.
Calculating Gradients
A gradient is a measure of steepness. The gradient is mathematically defined as the ratio of a functions rise over its run. Note: The letter is commonly used to refer to the gradient.
Gradient Between Two Points
Gradients To Angles
The angle between the positive x-axis and a straight line can be found by relating the angle to the gradient.
Gradient To Angle Formula
An explanation of this formula can be found below:
Angle Gradient Formula Explanation
We begin by applying the tan trigonometric ratio the the small triangle formed below the straight line.
Since we know we can write this as:
The Gradients Of Curves
We commonly talk about the gradient of straight lines, however we can also talk about the gradient of a curve at particular points. We can look at the steepness of the curve to give us an indication of the gradient.
Undefined Gradient Of Curves
The gradient of a smooth curve can be described at every point. However, if the curve has discontinuous or jagged points, the gradient cannot be found at those points. An example of each point is shown below:
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SOLUTION
Difficulty
2 Mins
Time
The graph of is shown below. Mark all points where the rate is undefined. Justify your reasoning for each location.
SOLUTION
The locations where calculus would struggle to find a rate are shown below:
Calculus would struggle at A because the graph is discontinuous at that point.
Calculus would struggle at B because there is a sudden change in the gradient. In other words, the function is jagged at that point.
Difficulty
4 Mins
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The graph below shows the speed of two cyclists during their daily commute.
Mark all the points on the graph where the cyclist’s speed is increasing, decreasing or stationary.
Describe each cyclists journey using terms such as increasing, decreasing, maximum and minimum.
SOLUTION
Part 1
The green highlights represent an increasing speed, the red highlights represent a decreasing speed and the orange highlights represent a stationary speed.
Part 2
Cyclist A’s journey begins at a lower speed of 2.5m/s, however it rapidly begins to increase until it reaches 5m/s at the 3min mark. From here the speed begins to decrease until it reaches a minimum speed of 2m/s at the 7.5min mark. At this point the speed begins to increase for the rest of her journey, however the rate of increase becomes much more gradual towards the end.
Cyclist B’s journey begins with a relatively high speed of 5.5m/s. At the start of his journey the speed is increasing until it reaches a maximum speed of 9m/s at around the 5.5min mark. From here the speed gradually decreases for the rest of the journey, however the rate at which it decreases becomes much more gradual towards the end of the journey.
Difficulty
5 Mins
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Water is being poured into a vase at a constant rate. The graph below shows the height of the water line during the pour.
Use the provided information to draw the cross-sectional view of the vase. Justify the shape in each region.
SOLUTION
To answer this question we must use some basic intuition that relates the rate at which the height increases ( gradient of the curve ) to the width of the vase at that height.
Intuition Breakdown
When the height is increasing rapidly it reasons that the radius of the vase must be quite small at that height
When the height is increasing at a constant rate it reasons that the radius of the vase is staying constant.
When the height is increasing slowly it reasons that the radius of the vase must be quite large at that height.
Approximate Shape
Using this reasoning we can come up with the cross-sectional diagram below:
Justification At Each Time
0 - 5: The gradient is small and increases rapidly. This corresponds to a large radius that transitions into a very small one.
5 - 6: The gradient goes from very large to moderately large in a small time. This corresponds to a small radius transitioning into a larger radius in short height range.
6 - 13: The gradient is quite constant during this region which corresponds to a constant radius
13 - 17: The gradient is quite small, meaning the radius is quite large causing a minimal increase in height.
Difficulty
3 Mins
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Mr Beast built a lego tower with a constant cross-sectional area. The plot below shows his progress over the 15 hours it took him to build it.
Which hour was his most efficient
Which hour was his least efficient
SOLUTION
The efficiency of the building can quantified by measuring how much height the tower gained in that hour. We are essentially looking to measure the following quantity:
Since this is a height vs time graph, we can see that the gradient of the curve represents our efficiency metric. This is because the rise is in meters and the run is in hours, making the gradient meters per hour.
The 5th hour was the most efficient as it has the largest gradient
the 10th hour was the least efficient as it has the smallest gradient
Difficulty
4 Mins
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The one tree hill circuit is a popular tourist hike in Melbourne Australia. The graph below shows the altitude of the hike vs the ground distance.
Find the gradient of the slope between the highest point on the hike and end of the hike ( both points are marked on the graph ).
SOLUTION
The slope gradient can be found by applying the gradient formula.
It is important to use the same units for both the rise and the run, hence all km measurements will be converted into m.
This tells us that we drop 59m of altitude for every 750m of ground distance we cover.
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Presets
Gradients and Calculus
6B - Gradients of intervals and lines
Question 2 (b, c, e)
Question 3 (a, d, e)
Question 6 (a, b)
Question 7 (b, c)
Question 9
Question 16 (a, b)
Question 21
Question 22
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What is Calculus
Review Questions
On a high level, calculus is the study of?
How quickly a value grows when it is repeatedly doubled
How to change functions into other similar functions
The effect of changing one value on the value of another
How different ratios on the circle change as we modify the angle
What kinds of functions does calculus have difficulty with?
Functions that change height very quickly
Functions that are undefined at particular places
Functions that you can draw without lifting your pen
Functions that suddenly change height and have sharp points
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What is Calculus
Review Questions
On a high level, calculus is the study of?
The effect of changing one value on the value of another
How to change functions into other similar functions
How quickly a value grows when it is repeatedly doubled
How different ratios on the circle change as we modify the angle
What kinds of functions does calculus have difficulty with?
Functions that suddenly change height and have sharp points
Functions that are undefined at particular places
Functions that change height very quickly
Functions that you can draw without lifting your pen
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Introducing Gradients
Review Questions
What is a gradient between two points
The change in the x values divided by the change in the y values
A measure of the similarity between the two points
The change in the y values divided by the change in the x values
A coordinate that stores the change in the x values and the change in the y values between the two points
How can we relate gradients to angles?
We can find the gradient from an angle with .
We can find the gradient from an angle with .
We can find the gradient from an angle with .
Gradients are not directly related to angles
If we compare a gradient of 1 to a gradient of 3, we can say that:
The angle when is one third of the angle at
That we move three times as far horizontally for the same vertical movement
That we rise three times as high for the same horizontal movement
The angle when is three times larger than the angle at
What is the difference between a positive and a negative gradient?
As we move to the right, positive gradients indicate upwards movement and negative gradients indicate downwards movement
As we move to the right, positive gradients indicate downwards movement and negative gradients indicate upwards movement
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Introducing Gradients
Review Questions
What is a gradient between two points
A coordinate that stores the change in the x values and the change in the y values between the two points
A measure of the similarity between the two points
The change in the x values divided by the change in the y values
The change in the y values divided by the change in the x values
How can we relate gradients to angles?
We can find the gradient from an angle with .
Gradients are not directly related to angles
We can find the gradient from an angle with .
We can find the gradient from an angle with .
If we compare a gradient of 1 to a gradient of 3, we can say that:
The angle when is one third of the angle at
That we move three times as far horizontally for the same vertical movement
That we rise three times as high for the same horizontal movement
The angle when is three times larger than the angle at
What is the difference between a positive and a negative gradient?
As we move to the right, positive gradients indicate upwards movement and negative gradients indicate downwards movement
As we move to the right, positive gradients indicate downwards movement and negative gradients indicate upwards movement