Applied Calculus — Unit 1–3 Important Questions (Part A): 22 flashcards
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1. Define the derivative of $f(x)$ using a limit.
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$$f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$$
2. What does the derivative represent geometrically?
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The slope of the tangent line to the curve at that point.
3. State the power rule for differentiation.
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$\frac{d}{dx}(x^n) = n x^{n-1}$
4. Differentiate $f(x) = x^4 + x^2$.
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$f'(x) = 4x^3 + 2x$
5. What is the first-order condition for an extremum at an interior point?
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$f'(x) = 0$ at that point (a stationary point).
6. State the second-order test for a local maximum.
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$f''(x) < 0$ at the stationary point.
7. State the second-order test for a local minimum.
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$f''(x) > 0$ at the stationary point.
8. What does a partial derivative $\partial f/\partial x$ measure?
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The rate of change of $f$ with respect to $x$ while holding all other variables constant.
9. State the chain rule for $f(x(t), y(t))$.
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$$\frac{df}{dt} = \frac{\partial f}{\partial x}\frac{dx}{dt} + \frac{\partial f}{\partial y}\frac{dy}{dt}$$
10. Define a function homogeneous of degree $n$.
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$f(\lambda x, \lambda y) = \lambda^n f(x, y)$ for every $\lambda > 0$.
11. State Euler's theorem for a homogeneous function of degree $n$.
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$$x\frac{\partial f}{\partial x} + y\frac{\partial f}{\partial y} = n\,f(x,y)$$
12. Write the Lagrangian used in the Lagrange multiplier method.
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$\mathcal{L}(x,y,\lambda) = f(x,y) - \lambda\, g(x,y)$
13. What are the stationary-point conditions for unconstrained optimisation of $f(x,y)$?
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$f_x = 0$ and $f_y = 0$ simultaneously.
14. What is an indefinite integral $\int f(x)\,dx$?
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The family of all antiderivatives $F(x) + C$ of $f$.
15. State the fundamental theorem of calculus.
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$$\int_a^b f(x)\,dx = F(b) - F(a), \text{ where } F' = f$$
16. What makes an integral improper?
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The interval is unbounded, or the integrand is unbounded somewhere in the interval.
17. When does an improper integral converge?
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When the defining limit of proper integrals exists (is finite).
18. State the reduction formula for $\int \sin^n x\,dx$.
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$$\int \sin^n x\,dx = -\frac{\sin^{n-1}x\cos x}{n} + \frac{n-1}{n}\int \sin^{n-2}x\,dx$$
19. What is the purpose of a reduction formula?
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To express an integral in terms of a lower power until a base case ($n=1$ or $n=2$) is reached.
20. If $f$ is position, what is $f'$?
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Velocity — the instantaneous rate of change of position.
21. If $f$ is cost, what is $f'$?
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Marginal cost — the rate at which cost changes with output.
22. What does the total derivative add over partial derivatives?
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It accounts for the inputs themselves changing with a common variable $t$, combining all effects via the chain rule.
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