AY2024 Sem 2 E0001F_L08_Lecture Notes (Student)
Learning Outcome
Ability to estimate the gradient of a curve by drawing a tangent.
Gradients of Curves (Tangent Lines)
Understanding Tangents
A point A on the curve has a tangent line AT that touches the curve at A.
The tangent is a straight line that represents the slope (gradient) at that specific point.
Gradient Definition
The gradient of a curve at a point is defined as the gradient of the tangent at that point.
Gradients can vary:
Positive
Zero
Negative
Visualization
Dotted lines represent tangents to the curve at specific points, while dashed lines are not tangents.
Worked Example 1: Calculating Gradient
Step-by-Step Task
Tabulate at least 6 points for the equation ( y = 8 - 2x^2 )
Range: (-3 \leq x \leq 3)
Anticipated values for y:
( x = -3 \rightarrow y = -10 )
( x = -2 \rightarrow y = 0 )
( x = -1 \rightarrow y = 6 )
( x = 0 \rightarrow y = 8 )
( x = 1 \rightarrow y = 6 )
( x = 2 \rightarrow y = 0 )
( x = 3 \rightarrow y = -10 )
Graphical Representation
Draw x and y axes for the range above.
Plot the points on the graph.
Connect the points with a flexible ruler to depict the curve.
Drawing the Tangent
Choose a point on the curve and draw a tangent line that only touches at that point.
Select two easy-to-identify points on this tangent.
Gradient Calculation
Use selected points to calculate the gradient of the tangent line.
Specific Points for Gradient Calculation
At (x = -2):
The gradient is calculated using points chosen from the tangent.
At (x = 0):
Tangent is horizontal; gradient = 0.
At (x = 1):
Gradient calculations involve identifying and substituting the right coordinates.
Summary Results
Gradient trends:
Positive for (x < 0)
Zero at (x = 0)
Negative for (x > 0)
Test Yourself (Exercise 1)
Tasks to Solve
Plot: ( y = 2x^2 + 1 ) for (-3 \leq x \leq 3)
Find gradients at:
(i) ( x = 0 )
(ii) ( x = 2 )
(iii) ( x = -1.5 )
Use the graph to find:
(iv) Value of (y) when (x = -0.5)
(v) Values of (x) when (y = 8)
(vi) Least value of (y) and corresponding (x).
Example Solution for Exercise 1
Tabulated Points
( x: -3, -2, -1, 0, 1, 2, 3 ) ( y: 19, 9, 3, 1, 3, 9, 19 )
Plotting and Connection
Graphing steps are similar to prior setups.
Working with Motion: Height of a Ball
Scenario Setup
Formula for height above a cliff: ( h = 25t - 5t^2)
Plot the graph using a 0.5 s intervals over 6 seconds.
Tasks
Find:
(i) Value of (h) at (t = 4) seconds
(ii) Greatest height above the cliff.
(iii) Gradients at (t = 1) and (t = 5) via tangent lines.
Results Summary
Evaluated Points
At (t = 4 \rightarrow h = 20 m )
Greatest height calculated to be ( 31.25 m )
Gradient Calculations
For height at ( t = 1 ) and ( t = 5 ) from tangent lines, apply derived mathematical approaches or visual estimation to compute results.
Final Observations
The calculated gradients reveal various properties of motion described.
Through exercises and applications, reinforce understanding of practical uses of gradients in mathematics.
Application of Graphs
Lesson Overview
Course: Foundational Mathematics
Institution: Republic Polytechnic
Focus: Application of Graphs (Lesson 08 E0001F)
Learning Outcome
Ability to estimate the gradient of a curve by drawing a tangent.
Gradients of Curves (Tangent Lines)
Understanding Tangents
A point A on the curve has a tangent line AT that touches the curve at A.
The tangent is a straight line that represents the slope (gradient) at that specific point.
Gradient Definition
The gradient of a curve at a point is defined as the gradient of the tangent at that point.
Gradients can vary:
Positive
Zero
Negative
Visualization
Dotted lines represent tangents to the curve at specific points, while dashed lines are not tangents.
Worked Example 1: Calculating Gradient
Step-by-Step Task
Tabulate at least 6 points for the equation ( y = 8 - 2x^2 )
Range: (-3 ≤ x ≤ 3)
Anticipated values for y:
( x = -3 → y = -10 )
( x = -2 → y = 0 )
( x = -1 → y = 6 )
( x = 0 → y = 8 )
( x = 1 → y = 6 )
( x = 2 → y = 0 )
( x = 3 → y = -10 )
Graphical Representation
Draw x and y axes for the range above.
Plot the points on the graph.
Connect the points with a flexible ruler to depict the curve.
Drawing the Tangent
Choose a point on the curve and draw a tangent line that only touches at that point.
Select two easy-to-identify points on this tangent.
Gradient Calculation
Use selected points to calculate the gradient of the tangent line.
Specific Points for Gradient Calculation
At (x = -2): The gradient is calculated using points chosen from the tangent.
At (x = 0): Tangent is horizontal; gradient = 0.
At (x = 1): Gradient calculations involve identifying and substituting the right coordinates.
Summary Results
Gradient trends:
Positive for (x < 0)
Zero at (x = 0)
Negative for (x > 0)
Test Yourself (Exercise 1)
Tasks to Solve
Plot: ( y = 2x^2 + 1 ) for (-3 ≤ x ≤ 3)
Find gradients at:
( x = 0 )
( x = 2 )
( x = -1.5 )
Use the graph to find:
Value of (y) when (x = -0.5)
Values of (x) when (y = 8)
Least value of (y) and corresponding (x).
Example Solution for Exercise 1
Tabulated Points
( x: -3, -2, -1, 0, 1, 2, 3 ) ( y: 19, 9, 3, 1, 3, 9, 19 )
Plotting and Connection
Graphing steps are similar to prior setups.
Working with Motion: Height of a Ball
Scenario Setup
Formula for height above a cliff: ( h = 25t - 5t^2)
Plot the graph using a 0.5 s intervals over 6 seconds.
Tasks
Find:
Value of (h) at (t = 4) seconds
Greatest height above the cliff.
Gradients at (t = 1) and (t = 5) via tangent lines.
Results Summary
Evaluated Points
At (t = 4 → h = 20 m )
Greatest height calculated to be ( 31.25 m )
Gradient Calculations
For height at ( t = 1 ) and ( t = 5 ) from tangent lines, apply derived mathematical approaches or visual estimation to compute results.
Final Observations
The calculated gradients reveal various properties of motion described.
Through exercises and applications, reinforce understanding of practical uses of gradients in mathematics.