Recent content by Joe20

  1. J

    Chemistry Predicting the electron domain and molecular geometries

    I would like to know how can we determine or tell from the diagram that the electron domain is tetrahedral and molecular geometry is bent/ v-shaped? thanks
  2. J

    Legendre polynomial - recurrence relations

    Note: $P_n (x)$ is legendre polynomial $$P_{n+1}(x) = (2n+1)P_n(x) + P'_{n-1}(x) $$ $$\implies P_{n+1}(x) = (2n+1)P_n(x) + \sum_{k=0}^{\lfloor\frac{n}{2}\rfloor} (2(n-1-2k)+1)P_{n-1-2k}(x))$$ How can I continue to use induction to prove this? Help appreciated.
  3. J

    MHB Can Rationalizing the Denominator Solve a Stuck Homogeneous Equation?

    Hi, I have solved this ODE till half way and got stucked on the integration of some weird expression. Need help for this. Thank you!
  4. J

    MHB How can I solve a non-homogeneous equation using substitution?

    Hi, I have attached part of my steps for solving the homogeneous equation. The equation is proven to be homogeneous. However after using substitution of y=zx and its' derivative, I was not able to separate the variables conveniently as shown. Please advise. Thank you!
  5. J

    MHB Using Rolle's theorem to prove for roots (part 2)

    Here is the revised proof: Proof: Let f(x) = ax^3 + bx + c. Assume that f(x) has 3 distinct roots, f(p) = f(q) = f(r) = 0, there is a point x1 element of (p,q) such that f'(x1) = 0 ; x2 element of (q, r) such that f'(x2) = 0. Since ab < 0 then there are two possibilities where a>0 and b<0...
  6. J

    MHB Using Rolle's theorem to prove for roots (part 2)

    Hi, I have done up the proof for the question below. Please correct me if I have done wrong for the proof. Thanks in advanced!Question: Prove that if ab < 0 then the equation ax^3 + bx + c = 0 has at most three real roots.Proof: Let f(x) = ax^3 + bx + c. Assume that f(x) has 4 distinct...
  7. J

    MHB Using Rolle's theorem to prove for roots

    Re: Using rolle's theorem to prove for roots Thanks, will change the usage of a and b accordingly.
  8. J

    MHB Using Rolle's theorem to prove for roots

    I have deduce a proof as stated below and am not sure if it is correct, therefore need some advice. Question: Prove that if ab > 0 then the equation ax^3 + bx + c = 0 has exactly one root by rolle's theoremProof: Let f(x) = ax^3+bx+c = 0. f(x) is continuous and differentiable since it is a...
  9. J

    MHB Continuous and differentiability

    Hello, I have attached the question and the steps worked out. I am not sure if my steps are correctly. Need advise on that. Next, I am not sure how to show f''(0) exist or not. Thanks in advance!
  10. J

    MHB Can I Find Examples for Continuous Improvement?

    Need help for the two parts! Thanks!
  11. J

    MHB Understanding the Cauchy-Schwarz Inequality

    I am not sure what examples to give, need help on this. Have attached the theorem as well.
  12. J

    MHB Find Limit of cos(x) with Inequalities | Part (b) Help

    Need advice on how to find lim of cos(x) using the inequalities provided. Also part (b) for help. Thanks.
  13. J

    MHB Is Showing One ε Enough to Prove Discontinuity?

    Hi, may I ask how do you know that f(U)⊆(1/2,3/2) and how do u know ε=1/2? How do I answer to the f(x) = 1 and 0? I am confused.
  14. J

    MHB Can Dividing by Sin x Help Prove Continuity at x = 0?

    Not sure how to do this question. Help needed. Thanks
  15. J

    MHB Is Showing One ε Enough to Prove Discontinuity?

    Appreciate the help needed for the attached question. Thanks!