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Introduction

Introduction. We will cover 3 topics today 1. Coursework Feedback 2 . Tangent Planes Recap 3 . Exam Questions. Coursework Feedback. Large values of x. As x → ∞ then p(x) → -∞. x → -∞ then p(x) → ∞. Show your working. i.e. not. Neatness. Coursework Feedback. y. Correct.

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Introduction

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  1. Introduction We will cover 3topics today 1. Coursework Feedback2. Tangent Planes Recap3. Exam Questions

  2. Coursework Feedback Large values of x As x → ∞ then p(x) → -∞ x → -∞ then p(x) → ∞ Show your working i.e. not Neatness

  3. Coursework Feedback y Correct f(x) → ∞ (0, 5) (1, 0) x f(x) → -∞

  4. Coursework Feedback y Incorrect x

  5. Coursework Feedback If you copy other people’s coursework then you are cheating. People caught copying one another will be given zero marks.

  6. Tangent Planes Recap Find the equation of the tangent plane at the point (2, 1, -2) on the sphere x2 + y2 + z2 = 9 We get the negative sign because the z-coordinate is negative Work out the coefficients first. The chain rule gives Therefore the equation of the tangent plane is

  7. Exam Questions Show that the following function has a root that lies between x = 0 & 1 Newton’s method for finding the root of the equation f(x) = 0 uses the following iterative scheme. n = 0, 1, 2… Starting from the point x0 = 0.5 obtain a root of the equation that is accurate to 2 decimal places. Show all of your working.

  8. Exam Questions Answer The change in sign indicates a root between x = 0 and x = 1. Using Newton-Raphson to find the root thus 0 0.5 1.218 6.437 −0.189 0.311 1 0.311 0.174 4.725 −0.037 0.274 2 0.274 0.004 4.460 −0.001 0.273 3 0.273

  9. Exam Questions The Maclaurin series expansion of the function f(x) is given (in the standard notation) by Find the Maclaurin series expansion of In ascending powers of x up to and including x4.

  10. Exam Questions Answer Differentiate using the product rule Hence

  11. Exam Questions A function of two variables f(x, y) is given by a) Calculate the first-order partial derivatives fx and fy. b) Calculate the second-order partial derivatives fxy and fyy and demonstrate that

  12. Exam Questions Answer To evaluate fx and fy use the product rule.

  13. Exam Questions Answer To evaluate fxy and fyy use the product rule again.

  14. Conclusion Essential reading for next week FINISH OFF YOUR HELM READING!!!!!! Today we have looked at 1. Coursework Feedback2. Tangent Planes Recap3. Exam Questions

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