Applied Calculus: 16 MCQs

All 16 multiple-choice questions from this kit, free and with no sign-up.

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Warm-up · 6 questions

Q1. Find the derivative of $f(x) = x^2$.

Beginner

A $x$

B $2x$

C $x^2$

D $2$

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Q2. Which of the following best defines the derivative of a function $f(x)$?

Beginner

A The area under the curve of $f$ between two points

B The slope of the tangent line to the graph of $f$ at a point

C The maximum value of $f$ on its domain

D The average value of $f$ over an interval

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Q3. Find the derivative of $f(x) = x^4 + x^2$.

Beginner

A $x^3 + x$

B $4x^4 + 2x^2$

C $4x^3 + 2x^2$

D $4x^3 + 2x$

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Q4. Which of the following correctly defines partial derivatives?

Beginner

A The integral of a function over a region

B The derivative of a function with respect to all variables simultaneously

C The derivative of a multivariable function with respect to one variable while holding others constant

D The limit of the function as it approaches infinity

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Q5. Which of the following best defines an indefinite integral?

Beginner

A The limit of a Riemann sum

B The definite integral over a closed interval

C The area under a curve between two points

D The family of all antiderivatives of a function, denoted $\int f(x)\,dx = F(x) + C$

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Q6. The derivative of a constant function is always zero.

Beginner

A True

B False

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Building up · 7 questions

Q7. State the conditions for maxima and minima of a function $f(x)$ at a critical point $c$.

Intermediate

A $f'(c) = 0$ and $f''(c) = 0$ always indicates an extremum

B $f'(c) = 0$ and $f''(c) > 0$ for a minimum, $f''(c) < 0$ for a maximum

C $f'(c)$ does not exist and $f''(c) > 0$ for a minimum

D $f'(c) > 0$ and $f''(c) > 0$ for a maximum

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Q8. State Euler's theorem for a homogeneous function $f(x,y)$ of degree $n$.

Intermediate

A $\frac{\partial f}{\partial x} + \frac{\partial f}{\partial y} = n f(x,y)$

B $x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y} = n f(x,y)$

C $x \frac{\partial f}{\partial x} - y \frac{\partial f}{\partial y} = n f(x,y)$

D $x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y} = f(x,y)$

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Q9. State the chain rule for a function $z = f(x,y)$ where $x = x(t)$ and $y = y(t)$.

Intermediate

A $\frac{dz}{dt} = \frac{\partial f}{\partial x} \frac{dy}{dt} + \frac{\partial f}{\partial y} \frac{dx}{dt}$

B $\frac{dz}{dt} = \frac{dx}{dt} + \frac{dy}{dt}$

C $\frac{dz}{dt} = \frac{\partial f}{\partial x} \frac{dx}{dt} + \frac{\partial f}{\partial y} \frac{dy}{dt}$

D $\frac{dz}{dt} = \frac{\partial f}{\partial x} + \frac{\partial f}{\partial y}$

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Q10. Which of the following best defines the total derivative of $z = f(x,y)$ with respect to $t$ when $x$ and $y$ are functions of $t$?

Intermediate

A $\frac{dz}{dt} = \frac{dx}{dt} \frac{dy}{dt}$

B $\frac{dz}{dt} = \frac{\partial f}{\partial x} + \frac{\partial f}{\partial y}$

C $\frac{dz}{dt} = \frac{\partial f}{\partial x} \frac{dy}{dt} + \frac{\partial f}{\partial y} \frac{dx}{dt}$

D $\frac{dz}{dt} = \frac{\partial f}{\partial x} \frac{dx}{dt} + \frac{\partial f}{\partial y} \frac{dy}{dt}$

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Q11. State the fundamental theorem of calculus.

Intermediate

A Every continuous function is differentiable

B The integral of a derivative is always zero

C The derivative of an integral is always zero

D If $f$ is continuous on $[a,b]$ and $F(x) = \int_a^x f(t)\,dt$, then $F'(x) = f(x)$

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Q12. Which of the following best defines an improper integral?

Intermediate

A An integral that cannot be evaluated

B An integral of a polynomial function

C An integral where the integrand becomes infinite or the interval of integration is infinite

D An integral over a finite interval with a continuous integrand

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Q13. Euler's theorem applies to any function, not just homogeneous functions.

Intermediate

A True

B False

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Topper level · 3 questions

Q14. State the conditions for maxima and minima of a function of two variables $f(x,y)$ at a critical point $(a,b)$.

Advanced

A $f_x = f_y = 0$ and $D < 0$ indicates a minimum

B $f_x = f_y = 0$ and $f_{xx} > 0$ indicates a maximum

C $f_x = f_y = 0$ and $D > 0$ always indicates a saddle point

D $f_x = f_y = 0$ and $D = f_{xx} f_{yy} - (f_{xy})^2 > 0$; if $f_{xx} > 0$ it's a minimum, if $f_{xx} < 0$ it's a maximum

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Q15. State Lagrange's multiplier method for finding extrema of $f(x,y)$ subject to the constraint $g(x,y)=0$.

Advanced

A Solve $\nabla f = 0$ and $\nabla g = 0$ separately

B Solve $\nabla f = \lambda$ and $g(x,y)=0$

C Solve $f(x,y) = g(x,y)$

D Solve $\nabla f = \lambda \nabla g$ together with $g(x,y)=0$

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Q16. Write the reduction formula for $\int \sin^n x \, dx$.

Advanced

A $\int \sin^n x \, dx = \frac{1}{n} \sin^{n-1} x \cos x + \frac{n-1}{n} \int \sin^{n-2} x \, dx$

B $\int \sin^n x \, dx = -\frac{1}{n} \sin^{n-1} x \cos x + \frac{n-1}{n} \int \sin^{n-2} x \, dx$

C $\int \sin^n x \, dx = -\frac{1}{n-1} \sin^{n-1} x \cos x + \frac{n-1}{n} \int \sin^{n-2} x \, dx$

D $\int \sin^n x \, dx = -\frac{1}{n} \sin^{n} x \cos x + \frac{n-1}{n} \int \sin^{n-1} x \, dx$

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