Scottish Higher Higher Mathematics Flashcards

24 sample cards from the Higher Mathematics deck

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

1. What does the derivative $f'(x)$ represent geometrically?

   It is the gradient of the tangent to the curve $y = f(x)$ at a given point, i.e. the instantaneous rate of change of $y$ with respect to $x$.

2. How do you find the stationary points of a curve and determine their nature?

   Solve $f'(x) = 0$ for the $x$-coordinates, then use a nature table of the sign of $f'(x)$ (or the second derivative) to classify each as a maximum, minimum, or point of inflection.

3. What does $f'(x) > 0$ and $f'(x) < 0$ tell you about a function?

   $f'(x) > 0$ means the function is increasing; $f'(x) < 0$ means it is decreasing on that interval.

4. What is the second derivative used for in curve analysis?

   $f''(x)$ gives concavity: $f''(x) > 0$ means concave up (minimum), $f''(x) < 0$ means concave down (maximum). It also measures the rate of change of the gradient.

5. State the rule for the indefinite integral $\int x^{n}\,dx$ where $n \neq -1$.

   $$\int x^{n}\,dx = \frac{x^{n+1}}{n+1} + C$$

6. State the integrals of $\sin x$ and $\cos x$.

   $\int \sin x\,dx = -\cos x + C$ and $\int \cos x\,dx = \sin x + C$ (with $x$ in radians).

7. State the Fundamental Theorem of Calculus for a definite integral.

   $$\int_a^b f(x)\,dx = F(b) - F(a)$$ where $F$ is an antiderivative of $f$.

8. How do you integrate $(ax+b)^{n}$ using the reverse chain rule?

   $$\int (ax+b)^{n}\,dx = \frac{(ax+b)^{n+1}}{a(n+1)} + C, \quad n \neq -1$$

9. How do you find the area between a curve $y = f(x)$ and the $x$-axis from $x=a$ to $x=b$?

   Compute $\int_a^b f(x)\,dx$. Where the curve is below the axis the integral is negative, so split the region at roots and take the modulus of each part.

10. How do you find the area enclosed between two curves $y = f(x)$ and $y = g(x)$?

    $$\text{Area} = \int_a^b \big(f(x) - g(x)\big)\,dx$$ where $f$ is the upper curve and $a$, $b$ are the $x$-coordinates of the intersection points.

11. What is the gradient of the line through points $(x_1,y_1)$ and $(x_2,y_2)$?

    $$m = \frac{y_2 - y_1}{x_2 - x_1}$$

12. State the relationship between gradients of perpendicular lines, and of parallel lines.

    Parallel lines have equal gradients $m_1 = m_2$. Perpendicular lines satisfy $m_1 m_2 = -1$.

13. State the point-gradient form of the equation of a straight line.

    $$y - b = m(x - a)$$ for a line of gradient $m$ through the point $(a, b)$.

14. What are the median, altitude, and perpendicular bisector of a triangle?

    A median joins a vertex to the midpoint of the opposite side; an altitude is perpendicular to a side through the opposite vertex; a perpendicular bisector passes through a side's midpoint at right angles to it.

15. State the equation of a circle with centre $(a,b)$ and radius $r$.

    $$(x - a)^{2} + (y - b)^{2} = r^{2}$$

16. For the general circle $x^{2}+y^{2}+2gx+2fy+c=0$, give the centre and radius.

    Centre $(-g, -f)$ and radius $r = \sqrt{g^{2}+f^{2}-c}$, valid when $g^{2}+f^{2}-c > 0$.

17. What is the relationship between a tangent to a circle and the radius at the point of contact?

    The tangent is perpendicular to the radius drawn to the point of contact, so the product of their gradients is $-1$.

18. How do you determine the magnitude of a 3D vector $\vec{u} = \begin{pmatrix} a \\ b \\ c \end{pmatrix}$?

    $$|\vec{u}| = \sqrt{a^{2} + b^{2} + c^{2}}$$

19. State the scalar (dot) product of $\vec{a}$ and $\vec{b}$ in component form and in terms of the angle between them.

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

20. How do you find the angle between two vectors, and what does $\vec{a}\cdot\vec{b}=0$ mean?

    $\cos\theta = \dfrac{\vec{a}\cdot\vec{b}}{|\vec{a}||\vec{b}|}$. If $\vec{a}\cdot\vec{b}=0$ (and neither is zero) the vectors are perpendicular.

21. State the section formula / condition for three points to be collinear using vectors.

    Points A, B, C are collinear if $\overrightarrow{AB} = k\,\overrightarrow{BC}$ for a scalar $k$ (the vectors are parallel and share point B), giving a common direction.

22. State the $n$th term and the recurrence relation form for a linear recurrence $u_{n+1} = a u_n + b$.

    Each term is found from the previous: $u_{n+1} = a u_n + b$, where $a$ is the multiplier and $b$ the added constant.

23. What is the condition for a linear recurrence relation $u_{n+1} = a u_n + b$ to have a limit?

    A limit exists if and only if $-1 < a < 1$ (i.e. $|a| < 1$).

24. State the formula for the limit $L$ of a convergent recurrence $u_{n+1} = a u_n + b$.

    $$L = \frac{b}{1 - a}, \quad |a| < 1$$ This is the value where $u_{n+1} = u_n = L$.

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