Scottish Advanced Higher Mathematics Flashcards

24 sample cards from the Mathematics deck

Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.

1. Give the derivatives of $\tan x$, $\sec x$, and $\ln x$.

   $\dfrac{d}{dx}\tan x = \sec^{2} x$; $\dfrac{d}{dx}\sec x = \sec x \tan x$; $\dfrac{d}{dx}\ln x = \dfrac{1}{x}$.

2. What are the derivatives of $\arcsin x$ and $\arctan x$?

   $$\frac{d}{dx}\arcsin x = \frac{1}{\sqrt{1 - x^{2}}}, \qquad \frac{d}{dx}\arctan x = \frac{1}{1 + x^{2}}.$$

3. How do you find $\frac{dy}{dx}$ by implicit differentiation?

   Differentiate every term with respect to $x$, applying the chain rule to functions of $y$ (so a term in $y$ gives $\frac{dy}{dx}$), then rearrange to solve for $\frac{dy}{dx}$.

4. For a curve given parametrically by $x(t)$ and $y(t)$, how do you find $\frac{dy}{dx}$?

   $$\frac{dy}{dx} = \frac{dy/dt}{dx/dt}, \quad \frac{dx}{dt} \neq 0.$$

5. How do you classify a stationary point using the second derivative test?

   At a stationary point ($f'(x)=0$): if $f''(x) > 0$ it is a minimum; if $f''(x) < 0$ it is a maximum; if $f''(x) = 0$ the test is inconclusive (examine the nature otherwise).

6. What conditions identify a point of inflection?

   The concavity changes there, so $f''(x) = 0$ and $f''$ changes sign across the point. (A horizontal point of inflection also has $f'(x) = 0$.)

7. Give the standard integrals of $\frac{1}{x}$ and $\frac{1}{a^{2}+x^{2}}$.

   $$\int \frac{1}{x}\,dx = \ln|x| + c, \qquad \int \frac{1}{a^{2}+x^{2}}\,dx = \frac{1}{a}\arctan\frac{x}{a} + c.$$

8. State the formula for integration by parts.

   $$\int u \, \frac{dv}{dx}\,dx = uv - \int v\,\frac{du}{dx}\,dx.$$

9. How do you integrate $\int \frac{f'(x)}{f(x)}\,dx$?

   $$\int \frac{f'(x)}{f(x)}\,dx = \ln|f(x)| + c.$$

10. What is the standard integral giving an $\arcsin$ result?

    $$\int \frac{1}{\sqrt{a^{2} - x^{2}}}\,dx = \arcsin\frac{x}{a} + c.$$

11. State the volume of revolution about the $x$-axis for $y = f(x)$ between $x=a$ and $x=b$.

    $$V = \pi \int_{a}^{b} y^{2}\,dx = \pi \int_{a}^{b} [f(x)]^{2}\,dx.$$

12. How do you find the area between two curves $y = f(x)$ and $y = g(x)$ on $[a,b]$ where $f \geq g$?

    $$A = \int_{a}^{b} \big(f(x) - g(x)\big)\,dx.$$

13. Define the modulus and argument of a complex number $z = x + iy$.

    Modulus $|z| = \sqrt{x^{2} + y^{2}}$; argument $\arg z = \theta$ where $\tan\theta = \dfrac{y}{x}$, taken in the correct quadrant (usually $-\pi < \theta \leq \pi$).

14. State De Moivre's theorem.

    For $z = r(\cos\theta + i\sin\theta)$ and integer $n$: $$z^{n} = r^{n}\big(\cos n\theta + i\sin n\theta\big).$$

15. How are the $n$ distinct $n$-th roots of a complex number $r(\cos\theta + i\sin\theta)$ found?

    $$z_k = r^{1/n}\left(\cos\frac{\theta + 2\pi k}{n} + i\sin\frac{\theta + 2\pi k}{n}\right), \quad k = 0, 1, \dots, n-1.$$ They lie equally spaced on a circle of radius $r^{1/n}$.

16. What is the complex conjugate of $z = x + iy$, and what are $z\bar{z}$ and $z + \bar{z}$?

    $\bar{z} = x - iy$; $z\bar{z} = x^{2} + y^{2} = |z|^{2}$; $z + \bar{z} = 2x = 2\,\mathrm{Re}(z)$.

17. Describe the method for solving a first order linear ODE $\frac{dy}{dx} + P(x)y = Q(x)$.

    Multiply by the integrating factor $I = e^{\int P\,dx}$, giving $\frac{d}{dx}(Iy) = IQ$. Then integrate: $y = \dfrac{1}{I}\int IQ\,dx$.

18. How do you solve a first order separable differential equation?

    Rearrange to $g(y)\,dy = f(x)\,dx$ (separating variables), then integrate both sides: $\int g(y)\,dy = \int f(x)\,dx + c$.

19. Write the auxiliary (characteristic) equation for $a y'' + b y' + c y = 0$ and the form of solution when its roots are real and distinct ($m_1, m_2$).

    Auxiliary equation: $am^{2} + bm + c = 0$. For distinct real roots: $y = A e^{m_1 x} + B e^{m_2 x}$.

20. Give the general solution of $a y'' + b y' + c y = 0$ when the auxiliary equation has a repeated root $m$, and when it has complex roots $p \pm qi$.

    Repeated root $m$: $y = (A + Bx)e^{mx}$. Complex roots $p \pm qi$: $y = e^{px}(A\cos qx + B\sin qx)$.

21. What is the general solution structure of a non-homogeneous second order linear ODE?

    $y = y_c + y_p$: the complementary function $y_c$ (general solution of the homogeneous equation) plus a particular integral $y_p$ (any solution of the full equation, found by trial).

22. How do you find the angle between two vectors $\vec{a}$ and $\vec{b}$ in three dimensions?

    $$\cos\theta = \frac{\vec{a}\cdot\vec{b}}{|\vec{a}||\vec{b}|}, \quad \vec{a}\cdot\vec{b} = a_1 b_1 + a_2 b_2 + a_3 b_3.$$

23. What does the vector (cross) product $\vec{a} \times \vec{b}$ give geometrically, and what is its magnitude?

    It gives a vector perpendicular to both $\vec{a}$ and $\vec{b}$ (direction by the right-hand rule). Its magnitude is $|\vec{a}\times\vec{b}| = |\vec{a}||\vec{b}|\sin\theta$, the area of the parallelogram they span.

24. Distinguish the conditions for a function to be even, odd, or periodic.

    Even: $f(-x) = f(x)$ (symmetric about the $y$-axis). Odd: $f(-x) = -f(x)$ (rotational symmetry about the origin). Periodic with period $T$: $f(x + T) = f(x)$ for all $x$.

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