Test of Mathematics for University Admission (TMUA) Calculus Flashcards

24 sample cards from the Calculus deck

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

1. When the second derivative test gives $f''(a) = 0$, what method can be used instead to classify the stationary point?

   The first derivative (sign) test: examine the sign of $f'(x)$ just to the left and right of $x=a$ to see how the gradient changes.

2. Using the first derivative test, what sign pattern of $f'(x)$ around $x=a$ indicates a local maximum?

   $f'(x)$ changes from positive (increasing) to negative (decreasing) as $x$ passes through $a$: $+ \to 0 \to -$.

3. Using the first derivative test, what sign pattern of $f'(x)$ indicates a local minimum?

   $f'(x)$ changes from negative to positive: $- \to 0 \to +$.

4. What does $f'(x) > 0$ tell you about the function on an interval?

   The function is increasing on that interval (the curve slopes upward).

5. What does $f'(x) < 0$ tell you about the function on an interval?

   The function is decreasing on that interval (the curve slopes downward).

6. Find and classify the stationary point of $y = x^{2} - 4x + 1$.

   $\dfrac{dy}{dx} = 2x - 4 = 0 \Rightarrow x = 2$, giving $y = -3$. Since $\dfrac{d^{2}y}{dx^{2}} = 2 > 0$, it is a local minimum at $(2, -3)$.

7. What does the concavity of a curve relate to in terms of the second derivative?

   $f''(x) > 0$ means the curve is concave up (convex); $f''(x) < 0$ means concave down. A change of concavity occurs at a point of inflection.

8. Define a point of inflection.

   A point where the curve changes concavity (from concave up to concave down or vice versa); $f''(x) = 0$ and changes sign there. It is stationary only if $f'(x)=0$ too.

9. Indefinite integration is the reverse of what operation?

   Differentiation — it is antidifferentiation, finding a function whose derivative is the given integrand.

10. State the rule for indefinite integration of $x^{n}$.

    $$\int x^{n}\,dx = \frac{x^{n+1}}{n+1} + c, \quad n \neq -1$$ where $c$ is the constant of integration.

11. Why must the constant of integration $c$ always be included in an indefinite integral?

    Because differentiating any constant gives $0$, so infinitely many functions differing by a constant share the same derivative; $c$ accounts for all of them.

12. Evaluate $\displaystyle\int (6x^{2} - 4x + 3)\,dx$.

    $2x^{3} - 2x^{2} + 3x + c.$

13. Evaluate $\displaystyle\int \sqrt{x}\,dx$.

    $\displaystyle\int x^{1/2}\,dx = \dfrac{x^{3/2}}{3/2} + c = \dfrac{2}{3}x^{3/2} + c.$

14. How do you find the particular constant $c$ when given a point through which the integral curve passes?

    Integrate to get the general solution, substitute the known point's coordinates to form an equation, and solve for $c$.

15. State the Fundamental Theorem of Calculus used to evaluate a definite integral.

    $$\int_{a}^{b} f(x)\,dx = \big[F(x)\big]_{a}^{b} = F(b) - F(a),$$ where $F'(x) = f(x)$.

16. What key difference distinguishes a definite integral from an indefinite integral?

    A definite integral has limits $a$ and $b$ and evaluates to a number; the constant $c$ cancels out. An indefinite integral yields a function (plus $c$).

17. Evaluate $\displaystyle\int_{1}^{3} 2x\,dx$.

    $\big[x^{2}\big]_{1}^{3} = 9 - 1 = 8.$

18. How is the area under the curve $y = f(x)$ between $x = a$ and $x = b$ (with $f(x)\geq 0$) found?

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

19. What does a definite integral give when the curve lies below the $x$-axis over the interval?

    A negative value. The actual area is the modulus (absolute value) of that integral; split the integral at the roots to handle regions above and below separately.

20. How do you find the area enclosed between two curves $y = f(x)$ (upper) and $y = g(x)$ (lower) from $x=a$ to $x=b$?

    $$A = \int_{a}^{b} \big(f(x) - g(x)\big)\,dx,$$ integrating the upper curve minus the lower curve.

21. Outline the general method for solving an optimisation problem using calculus.

    Express the quantity to optimise as a function of one variable (using any constraint to eliminate others), differentiate and set the derivative to zero, solve for the variable, then use the second derivative to confirm a maximum or minimum.

22. In the context of rates of change, what does $\dfrac{dV}{dt}$ represent?

    The rate of change of volume $V$ with respect to time $t$ (how fast the volume is changing per unit time).

23. State the chain rule connection used to relate two rates of change, e.g. $\dfrac{dV}{dt}$ via $\dfrac{dV}{dr}$.

    $$\frac{dV}{dt} = \frac{dV}{dr} \times \frac{dr}{dt}.$$

24. How do you find the gradient of a graph at a point, and how is this connected to the area under its gradient function?

    The gradient at a point is the derivative $f'(x)$; conversely, the area under the gradient function $f'(x)$ between $a$ and $b$ recovers the change in $f$: $\int_a^b f'(x)\,dx = f(b)-f(a)$. Differentiation and integration are inverse processes connecting a graph and its gradient/area.

What this deck covers

Calculus flashcards FAQ