WMA13 Pure Mathematics 3

Lessons in WMA13

1. Differentiating e^x and ln x
   d/dx e^kx = ke^kx; d/dx ln x = 1/x.
2. Algebraic fractions and division
   Factorise fully before cancelling factors; divide by the reciprocal; use a common denominator to add; divide polynomials to split an improper fraction.
3. Functions: domain, range and composition
   fg(x) = f(g(x)), so do g first. Give a range in terms of f(x); find it from a sketch or by completing the square.
4. Inverse functions
   Rearrange y = f(x) for x and swap letters. The domain of f^-1 is the range of f; the graphs reflect in y = x.
5. The modulus function and its graphs
   |f(x)|: reflect the parts below the x-axis upwards. f(|x|): keep x ≥ 0 and reflect it in the y-axis.
6. Modulus equations and inequalities
   Solve f(x) = g(x) and -f(x) = g(x), then check. |x - a| < b gives one interval; |x - a| > b gives two.
7. Combining transformations of graphs
   Inside changes act on x in the opposite way to what you might expect; outside changes act on y directly. Move key points one transformation at a time.
8. Sec, cosec and cot
   sec = 1/cos, cosec = 1/sin, cot = 1/tan. 1 + tan² ≡ sec² and 1 + cot² ≡ cosec².
9. Inverse trigonometric functions
   arcsin and arctan give values from -π/2 to π/2; arccos gives values from 0 to π. Each graph reflects the restricted trig graph in y = x.
10. Compound angle formulae
    sin(A ± B) = sin A cos B ± cos A sin B; cos(A ± B) = cos A cos B ∓ sin A sin B; tan(A ± B) = (tan A ± tan B)/(1 ∓ tan A tan B).
11. Double angle formulae
    sin 2A = 2sin A cos A; cos 2A = cos² A - sin² A = 2cos² A - 1 = 1 - 2sin² A; tan 2A = 2tan A/(1 - tan² A).
12. The R addition form
    R = sqrt(a² + b²). Compare coefficients to get R cos α and R sin α, then tan α = (R sin α)/(R cos α). The maximum is R and the minimum is -R.
13. e^x and ln x: graphs and equations
    ln and e undo each other: e^ln x = x and ln(e^x) = x. y = e^x + c has asymptote y = c; ln x needs a positive input.
14. Using log graphs to find a model
    y = axⁿ: log y against log x (gradient n). y = kb^x: log y against x (gradient log b). The intercept gives log a or log k.
15. Exponential growth and decay
    In N = Ae^kt, A is the start value and k the growth rate (negative for decay). Use ln to find t or k. d/dx (a^x) = a^x ln a.
16. Differentiating sin, cos and tan
    sin kx gives k cos kx; cos kx gives -k sin kx; tan kx gives k sec² kx. Always work in radians.
17. The chain rule
    dy/dx = dy/du × du/dx: differentiate the outside, then multiply by the derivative of the inside. dy/dx = 1/(dx/dy).
18. The product rule
    y = uv gives dy/dx = u v' + v u'. Write u, v, u', v' first and factorise the answer.
19. The quotient rule and sec, cosec, cot
    y = u/v gives (v u' - u v')/v². sec x gives sec x tan x; cosec x gives -cosec x cot x; cot x gives -cosec² x.
20. Integrating e^x, 1/x and trig functions
    Divide by k: ∫ e^kx = e^kx/k, ∫ cos kx = sin kx/k, ∫ sin kx = -cos kx/k. ∫ 1/x dx = ln|x|. Add + c.
21. Integration by recognition
    f'(x)/f(x) integrates to ln|f(x)|; f'(x)(f(x))ⁿ integrates to (f(x))ⁿ + 1/(n + 1). Guess, differentiate, adjust.
22. Integration using trig identities
    Rewrite sin², cos² and tan² with double angle or Pythagorean identities before integrating.
23. Locating roots and iteration
    Continuous plus a sign change means a root. Iterate x_n+1 = g(x_n) and test the bounds (for example 2.2785 and 2.2795) to confirm accuracy.

Next exams for WMA13

• January 2027: Fri 15 Jan 2027, afternoon, 1h 30m
• Summer 2027: Tue 25 May 2027, afternoon, 1h 30m

From Pearson's final timetables. All exam dates. Always confirm your entries with your exams officer.