Applied Calculus — Unit 1–3 Important Questions (Part A): 18 MCQs

All 18 multiple-choice questions from this kit, free and with no sign-up.

Full kit · everything Flashcards · 22 Quick Revision · 16

Warm-up · 8 questions

Q1. Which expression is the limit definition of the derivative of $f(x)$?

Beginner

A $\lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$

B $\lim_{h \to \infty} \frac{f(x+h)-f(x)}{h}$

C $\frac{f(x+h)-f(x)}{h}$ without any limit

D $\lim_{h \to 0} \frac{f(x+h)+f(x)}{h}$

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Q2. A curve's tangent line at $x = 3$ has slope 7. What is $f'(3)$?

Beginner

A Cannot be determined from the tangent

B 7

C 21

D 3

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Q3. Using the power rule, what is $\frac{d}{dx}(x^2)$?

Beginner

A $x$

B $x^3/3$

C $2x^2$

D $2x$

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Q4. Differentiate $f(x) = x^4 + x^2$.

Beginner

A $4x^3 + 2x$

B $x^3 + x$

C $4x^3 + x$

D $4x^4 + 2x^2$

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Q5. At an interior extremum of a differentiable function, which condition must hold?

Beginner

A $f(x) = 0$

B $f'(x) > 0$

C $f'(x) = 0$

D $f''(x) = 0$

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Q6. At a stationary point, $f''(x) < 0$. What does this tell you?

Beginner

A The point is a local minimum

B The point is a local maximum

C The function has no extremum there

D The test is inconclusive

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Q7. What does the partial derivative $\partial f/\partial x$ measure?

Beginner

A Change in $f$ as all variables vary together

B The total change in $f$ along a path in $t$

C Change in $f$ as $x$ varies, with other variables held constant

D The value of $f$ when $x = 0$

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Q8. The fundamental theorem of calculus connects differentiation and integration: $\int_a^b f(x)\,dx = F(b) - F(a)$ where $F$ is any antiderivative of $f$.

Beginner

A True

B False

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Building up · 5 questions

Q9. A firm's output $Q(L, K)$ depends on labour and capital. Which derivative tells you the effect of extra labour alone?

Intermediate

A $dQ/dt$

B $\partial Q/\partial L$

C $\partial^2 Q/\partial L \partial K$

D $\partial Q/\partial K$

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Q10. For $f(x,y)$ with $x = x(t)$ and $y = y(t)$, which formula gives $df/dt$?

Intermediate

A $f_x \frac{dy}{dt} + f_y \frac{dx}{dt}$

B $f_x + f_y$

C $f_x \frac{dx}{dt} - f_y \frac{dy}{dt}$

D $f_x \frac{dx}{dt} + f_y \frac{dy}{dt}$

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Q11. For a homogeneous function of degree $n$, Euler's theorem states that:

Intermediate

A $x f_x + y f_y = f$

B $x f_x + y f_y = n f$

C $f_x + f_y = n f$

D $x f_x - y f_y = n f$

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Q12. In the Lagrange multiplier method for optimising $f$ subject to $g = 0$, the Lagrangian is:

Intermediate

A $\mathcal{L} = f \cdot g$

B $\mathcal{L} = f + g$

C $\mathcal{L} = f - \lambda g$

D $\mathcal{L} = \lambda f - g$

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Q13. What is $\int f(x)\,dx$?

Intermediate

A A single antiderivative $F(x)$

B The area under $f$ from $a$ to $b$

C The derivative of $f(x)$

D The family of antiderivatives $F(x) + C$

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Topper level · 5 questions

Q14. Which integral is improper?

Advanced

A $\int_1^{\infty} \frac{1}{x^2}\,dx$

B $\int_0^{\pi} \sin x\,dx$

C $\int_1^2 (x+1)\,dx$

D $\int_0^1 x^2\,dx$

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Q15. An improper integral is defined as a limit of proper integrals. When does it converge?

Advanced

A When the defining limit exists and is finite

B When the integrand tends to zero at infinity

C When the antiderivative is a polynomial

D When the interval is finite

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Q16. What is the main purpose of the reduction formula for $\int \sin^n x\,dx$?

Advanced

A Convert the sine integral into a cosine integral permanently

B Express the integral in terms of a lower power until a base case is reached

C Evaluate the integral without any further integration

D Replace the integrand by its Taylor series

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Q17. Given the function $f(x, y) = x^2y + xy^2$, find the total derivative $\frac{df}{dt}$ if $x = t^2$ and $y = 3t$.

Advanced

A $\frac{df}{dt} = 6t^2 + 6t^3$

B $\frac{df}{dt} = 2t(3t^2 + 3t) = 6t^3 + 6t^2$

C $\frac{df}{dt} = 3t(2t + 3)$

D $\frac{df}{dt} = 3t^2 + 2t^3$

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Q18. Using the Lagrange multiplier method, determine the maximum value of $f(x, y) = 3x + 4y$ subject to the constraint $g(x, y) = x^2 + y^2 - 25 = 0$. What are the coordinates of the maximum point?

Advanced

A Maximum at $(0, 5)$ with value 20

B Maximum at $(5, 0)$ with value 15

C Maximum at $(3, 4)$ with value 24

D Maximum at $(4, 3)$ with value 25

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