Calculus AND Basic Maths flash cards

1. 1.What is a function $y=f(x)$?

   A rule assigning to each input $x$ (from a domain) exactly one output $y$ (in the range). In physics, $x$ is often the independent variable (like time) and $y$ the dependent one.

   Hint: One input, one output.

2. 2.Distinguish domain and range of a function.

   Domain = set of allowed inputs $x$; Range = set of resulting outputs $y=f(x)$. E.g. for $y=\sqrt{x}$, domain is $x\ge 0$, range is $y\ge 0$.

   Hint: Inputs vs outputs.

3. 3.What does the graph of a linear function $y=mx+c$ look like, and what do $m,c$ mean?

   A straight line; $m$ is the slope (steepness) and $c$ is the $y$-intercept (value of $y$ at $x=0$).

   Hint: Slope-intercept form.

4. 4.How do you find the slope of a straight line through two points $(x_1,y_1),(x_2,y_2)$?

   $m=\dfrac{y_2-y_1}{x_2-x_1}=\dfrac{\Delta y}{\Delta x}$. It is the rise over run.

   Hint: Rise over run.

5. 5.What is the shape of $y=x^2$ and $y=ax^2+bx+c$?

   A parabola. $y=x^2$ opens upward with vertex at origin; general quadratic opens up if $a>0$, down if $a<0$, with vertex at $x=-\dfrac{b}{2a}$.

   Hint: Parabola, sign of $a$.

6. 6.Sketch behaviour of the rectangular hyperbola $y=\dfrac{k}{x}$ (with $k>0$).

   Two branches in the first and third quadrants; $y\to\infty$ as $x\to 0^+$ and $y\to 0$ as $x\to\infty$. Asymptotes are the axes. Models inverse proportion ($y\propto 1/x$).

   Hint: Inverse proportion.

7. 7.What do the graphs of $y=e^{x}$ and $y=e^{-x}$ look like?

   $y=e^{x}$ rises steeply, always positive, passes through $(0,1)$ (exponential growth). $y=e^{-x}$ decays from $(0,1)$ towards $0$ (exponential decay).

   Hint: Growth vs decay, both pass $(0,1)$.

8. 8.Describe the graph of $y=\sin x$ over one period.

   A smooth wave oscillating between $-1$ and $+1$, period $2\pi$, starting at $0$, peak $+1$ at $\pi/2$, back to $0$ at $\pi$, trough $-1$ at $3\pi/2$.

   Hint: Wave, amplitude 1, period $2\pi$.

9. 9.How does the graph of $y=A\sin(\omega x)$ differ from $y=\sin x$?

   Amplitude scales to $A$ (oscillates between $\pm A$) and period changes to $\dfrac{2\pi}{\omega}$. Larger $\omega$ means faster oscillation.

   Hint: $A$ sets height, $\omega$ sets period.

10. 10.What is the effect of $y=f(x)+c$ and $y=f(x-a)$ on a graph?

    $f(x)+c$ shifts the graph vertically up by $c$; $f(x-a)$ shifts it horizontally to the right by $a$. These are graph translations.

    Hint: Vertical vs horizontal shifts.

11. 11.Define radian measure of an angle.

    Angle (in radians) $=\dfrac{\text{arc length}}{\text{radius}}=\dfrac{s}{r}$. A full circle is $2\pi$ rad $=360^\circ$, so $1\text{ rad}\approx 57.3^\circ$.

    Hint: Arc over radius.

12. 12.Convert between degrees and radians.

    $\text{radians}=\text{degrees}\times\dfrac{\pi}{180}$. E.g. $90^\circ=\dfrac{\pi}{2}$, $60^\circ=\dfrac{\pi}{3}$, $30^\circ=\dfrac{\pi}{6}$.

    Hint: Multiply by $\pi/180$.

13. 13.State the definitions of $\sin\theta,\cos\theta,\tan\theta$ in a right triangle.

    $\sin\theta=\dfrac{\text{opp}}{\text{hyp}}$, $\cos\theta=\dfrac{\text{adj}}{\text{hyp}}$, $\tan\theta=\dfrac{\text{opp}}{\text{adj}}=\dfrac{\sin\theta}{\cos\theta}$.

    Hint: SOH-CAH-TOA.

14. 14.Give the values of $\sin$ and $\cos$ at $0,30,45,60,90$ degrees.

    $\sin: 0,\tfrac12,\tfrac{1}{\sqrt2},\tfrac{\sqrt3}{2},1$. $\cos: 1,\tfrac{\sqrt3}{2},\tfrac{1}{\sqrt2},\tfrac12,0$. (cos is sin reversed.)

    Hint: Standard-angle table.

15. 15.State the fundamental identity linking $\sin$ and $\cos$.

    $\sin^2\theta+\cos^2\theta=1$. Dividing gives $1+\tan^2\theta=\sec^2\theta$ and $1+\cot^2\theta=\csc^2\theta$.

    Hint: Pythagorean identity.

16. 16.Write the sine and cosine addition formulas.

    $\sin(A\pm B)=\sin A\cos B\pm\cos A\sin B$; $\cos(A\pm B)=\cos A\cos B\mp\sin A\sin B$.

    Hint: Sine mixes, cosine matches (with sign flip).

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