Increasing and Decreasing Functions Through Graphs and Differentiation
Explore how to determine whether a function is increasing or decreasing by analyzing graphs and using differentiation. Practice finding values of x for which a function is increasing or decreasing, and understand the concept of stationary points. Enhance your grasp of gradients and their significance in analyzing functions.
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Presentation Transcript
Increasing and decreasing functions. Consider the graph of ? = ?2
To determine whether the value of a function is increasing or decreasing, we use differentiation to evaluate the gradient. ? ? = ?2 EG:
Find the values of x for which f(x) is an increasing function: 1) ? ? = 3?2+ 8? + 2 2) ? ? = 4? 3?2 3) ? ? = 5 8? 2?2 Find the values of x for which f(x) is decreasing: 4) ? ? = ?2 9? If you finish quickly, sketch the graph of ? = ?(?). What does your answer tell you about the graph?
A function may have more than one set of values for which it is increasing or decreasing. Consider the function ? ? = ?3+ 3?2
1) ? ? = 2?3 15?2+ 36? increasing? 2) ? ? = 2?3 3?2 12? increasing? 3) ? ? = ?3 27? + 1 decreasing? 4) ? ? = 3?4 20?3+ 36?2+ 12 increasing?
Starter Increasing and Decreasing Functions Find the set of values for ?: 1) ? ? = 2?2+ 8? 11 decreasing? ? < ? 2) ? ? = 2?3+ 3?2 12? + 10 decreasing? ? < ? < ? 3) ? ? = ?3 3?2 24? 9 increasing? ? < ? and ? > ?
Activity: Gradients Susan Wall STEM website (login required) https://www.stem.org.uk/resources/elibrary/resource/34073/propertie s-functions
Stationary Points Stationary points occur when the gradient of a function is zero. Consider again the function ? ? = ?3+ 3?2
Practice questions Find the coordinates of the stationary points on each curve: 1) ? = ?3 ?2 ? + 1 2) ? = ?(?2 4? 3) 3) ? = ?2+54 ? 1 2(? 6) 4) ? = ?
To determine the nature of stationary/turning points, we use the second derivative. At the stationary point : ?2? ??2< 0 ? ?? ? ????? ??????? ?2? ??2> 0 ? ?? ? ????? ??????? (?2? ??2= 0 ????3? ??3 0 ?? ? ????? ?? ?????????)
Use calculus to find the coordinates of the stationary points of each curve. Determine the nature of the stationary points. 1) ? = ?3 3?2 24? 2) ? = 3?2+ 45? 75 ?3 3) ? = 2?3 24? 4) ? = ? 22(? 1)
Use calculus to find the coordinates of the stationary points of each curve. Determine the nature of the stationary points. 1)? = ?3 3?2 24? 4, 80 ???????, 2,28 ??????? 2)? = 3?2+ 45? 75 ?3 5,100 ???????, 3, 156 ??????? 3)? = 2?3 24? 2, 32 ???????, 2,32 ??????? 4 3, 4 27 ???????, 4)? = ? 22(? 1) 2,0 ???????
Activity: Floppy Hair Underground Maths https://undergroundmathematics.org/calculus-of-powers/floppy-hair
