Applied Calculus: Differential, Multivariable, and Integral Calculus
8 minFull explanation, worked examples & memory tricks in the app
Kit Library / Mathematics / Calculus
⚡ Topic Learning KitApplied Calculus
English 16 leveled MCQs Free
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All 16 questions, laddered from warm-up to topper-level, each with its answer and the reasoning behind it.
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
Answers and explanations are in the Veda app.
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
$x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y} = f(x,y)$
Answers and explanations are in the Veda app.
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
$\frac{dz}{dt} = \frac{\partial f}{\partial x} + \frac{\partial f}{\partial y}$
Answers and explanations are in the Veda app.
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
$\frac{dz}{dt} = \frac{\partial f}{\partial x} \frac{dx}{dt} + \frac{\partial f}{\partial y} \frac{dy}{dt}$
Answers and explanations are in the Veda app.
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 $F'(x) = f(x)$
Answers and explanations are in the Veda app.
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
Answers and explanations are in the Veda app.
Q13. Euler's theorem applies to any function, not just homogeneous functions.
Intermediate
A
True
B
False
Answers and explanations are in the Veda app.
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
Answers and explanations are in the Veda app.
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
$\int \sin^n x \, dx = -\frac{1}{n} \sin^{n} x \cos x + \frac{n-1}{n} \int \sin^{n-1} x \, dx$
Answers and explanations are in the Veda app.
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