PIL import Image

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img_path = "/mnt/data/17326893603953s7d117m.jpg" img = Image.open(img_path) img.show()It appears that I have opened the image successfully. Let me now analyze its contents to assist you with the problem. I'll extract any relevant mathematical details shortly. pytesseract

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textThe image contains a problem asking for the Big-O estimates of several functions. Here’s a breakdown of the tasks:

Q5 [15 marks]

You are asked to give a Big-O estimate for the following functions:

(a) $3n + 3^3 + 4$

(b) $1 + 2434 \times n$

(c) $\log_{10}(2^n) + 10! \times n^2$

I will now calculate the Big-O for each of these.

Solution:

(a) $3n + 3^3 + 4$

• Simplification: $3^3 = 27$, so the expression becomes: $3n + 27 + 4 = 3n + 31$
• Big-O Estimate: The highest degree term is $3n$, so the Big-O estimate is: $O(n)$

(b) $1 + 2434 \times n$

• Simplification: The constant 1 becomes insignificant as $n$ grows large, so the expression simplifies to: $2434n$
• Big-O Estimate: The highest degree term is $2434n$, so the Big-O estimate is: $O(n)$

(c) $\log_{10}(2^n) + 10! \times n^2$

• Simplification:

  • $\log_{10}(2^n)$ can be simplified using the logarithmic property $\log_{b}(x^y) = y \log_b(x)$: $\log_{10}(2^n) = n \log_{10}(2)$ The logarith of 2 is a constant, so this term simplifies to $O(n)$.
  • $10! \times n^2$ is a constant multiplied by $n^2$, so it simplifies to $O(n^2)$.

• Big-O Estimate: The highest degree term is $10! \times n^2$, so the Big-O estimate is: $O(n^2)$

Final Answers:

• (a) $O(n)$
• (b) $O(n)$
• (c) $O(n^2)$

Would you like further explanations or clarifications on any of these steps? Here are some related questions for you:

1. What is the difference between Big-O and Big-Theta notation?
2. How do we handle constants when simplifying a function for Big-O analysis?
3. What is the importance of the highest degree term in Big-O estimation?
4. Could there be cases where logarithmic terms impact Big-O estimates significantly?
5. Can we combine different types of functions (e.g., polynomial and logarithmic) in Big-O notation?

Tip: When calculating Big-O for polynomials and logarithmic functions, always focus on the highest degree term, as this dominates the behavior for large inputs.