Lesson Worksheet: Iterated Functions Mathematics

In this worksheet, we will practice using an iterative formula to approximate the roots of an equation.

Q1:

Given an initial value 𝑥, we can iterate a function using the recursive formula 𝑥=𝑓(𝑥).

Find 𝑥 given the function 𝑓(𝑥)=3𝑥2 with initial value 𝑥=2.

Q2:

We can define a sequence 𝑥,𝑥,𝑥, using the recursive formula 𝑥=𝑓(𝑥) for some function 𝑓 and initial value 𝑥.

Write the first 5 terms of the sequence generated by 𝑓(𝑥)=83𝑥, 𝑥=11.

  • A11,8,16,56,160
  • B11,41,115,337,1,019
  • C11,41,115,353,1,051
  • D11,25,83,241,731
  • E11,41,115,353,1,051

Q3:

We can iterate a function 𝑓(𝑥)=𝑥2+1𝑥 using the recursive formula 𝑥=𝑓(𝑥). Find the positive fixed point.

  • A22
  • B1
  • C12
  • D2
  • E2

Q4:

We can iterate the function 𝑓(𝑥)=|52𝑥| using the recursive formula 𝑥=𝑓(𝑥) with initial value 𝑥=8. Find 𝑥.

Q5:

We can iterate the function 𝑓(𝑥)=𝑥+𝑥+1 using the recursive formula 𝑥=𝑓(𝑥) with initial value 𝑥=0. Find 𝑥.

Q6:

True or False:

The recursive formula 𝑥=𝑓(𝑥) of the function 𝑓(𝑥)=𝑥+1,𝑥,𝑥+2,𝑥isevenisodd can produce an even number with initial value 𝑥=1.

  • AFalse
  • BTrue

Q7:

Michael saves $100 every month. The savings of the current month can be calculated with the recursive function 𝑠=𝑠+100, where 𝑠 is the savings of the previous month. Find the savings of the 4th month given that his initial savings were $1,000.

Q8:

We can iterate the function 𝑓(𝑥)=(2)log using the recursive formula 𝑥=𝑓(𝑥) with initial value 𝑥=10. Find 𝑥.

Q9:

We can iterate the function 𝑓(𝑥)=𝑒 using the recursive formula 𝑥=𝑓(𝑥) with initial value 𝑥=10. Find 𝑥 to the nearest two decimal places.

Q10:

We can iterate the function 𝑓(𝑥)=𝑥+3,𝑥,𝑥+2,𝑥,isprimeisnotprime using the recursive formula 𝑥=𝑓(𝑥) with initial value 𝑥=11. Find 𝑥.

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