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Calculus Quizzes, Trivia and Puzzles
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Calculus Trivia

Calculus Trivia Quizzes

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Differential or integral - whichever your preference, there is bound to be a quiz here which can test your grasp of the concepts of calculus.
8 Calculus quizzes and 75 Calculus trivia questions.
1.
No Calculus Knowledge Needed
  No Calculus Knowledge Needed!   top quiz  
Photo Quiz
 10 Qns
Throw away your instinctive fear of the words "math" and "calculus" as we explore the intuition and common sense behind calculus in a way that requires no calculation and no previous calculus knowledge. (Remember: you can click on images to zoom in!)
Average, 10 Qns, AdamM7, Jan 29 22
Average
AdamM7 gold member
Jan 29 22
244 plays
2.
  The Wonderful World of Differentiation   top quiz  
Multiple Choice
 10 Qns
Differentiation is a function found in the wonderful branch of mathematics, calculus! Can you differentiate? Enjoy! (Best played in interactive mode for players less familiar with calculus).
Average, 10 Qns, jonnowales, Jun 29 12
Average
jonnowales gold member
2470 plays
3.
  Quest for Calculus   popular trivia quiz  
Multiple Choice
 10 Qns
This quiz will cover your knowledge of basic differential and integral calculus. Drop me a line if you like this quiz!
Tough, 10 Qns, Diamondlance, May 20 17
Tough
Diamondlance gold member
May 20 17
6055 plays
4.
  You Derive Me Crazy!   great trivia quiz  
Multiple Choice
 10 Qns
Let's explore a few of the concepts related to differential calculus. It's not as scary as it sounds!
Average, 10 Qns, looney_tunes, May 10 15
Average
looney_tunes editor
578 plays
5.
  The Tangent Line    
Multiple Choice
 10 Qns
The concept of tangent line is fundamental to a first course in calculus. Here are ten questions for calculus lovers involving tangent lines. You will need a lot of time for this one! Good luck!
Tough, 10 Qns, rodney_indy, Jan 29 22
Tough
rodney_indy
Jan 29 22
776 plays
6.
  Small Derivatives Quiz    
Multiple Choice
 5 Qns
This is an easy quiz for anyone who took calculus. Good Luck!
Average, 5 Qns, Lanire, Nov 08 21
Average
Lanire
Nov 08 21
8374 plays
7.
  Calculus Fundamentals    
Multiple Choice
 10 Qns
There's a lot more to Calculus than just integrating and differentiating mindlessly. This quiz will cover some of the theorems and concepts behind the area. Some questions are hard, but you shouldn't need to do any actual computations.
Very Difficult, 10 Qns, kevinatilusa, Mar 26 03
Very Difficult
kevinatilusa
3705 plays
8.
  Calculus Fun (or My Head Hurts!)    
Multiple Choice
 10 Qns
This quiz is based on infinitesimal calculus (specifically single-variable). It involves the history, definitions, and some problems to solve. Good luck!
Average, 10 Qns, hausc018, Jun 29 12
Average
hausc018
759 plays

Calculus Trivia Questions

1. One of the uses of differential calculus is describing how steeply a curve is rising or falling. This is measured by a straight line which touches the curve at exactly one point. What is the name for such a line?

From Quiz
You Derive Me Crazy!

Answer: Tangent

The word tangent may be familiar from its use to name one of the three trigonometric functions commonly met in high school (along with the sine and the cosine). In geometry, it means a line or curve which touches another one at exactly one point. The word comes from the Latin verb 'tangere', to touch.

2. What is the broadest definition of calculus?

From Quiz Calculus Fun (or My Head Hurts!)

Answer: the study of change

Calculus involves many other components when defining it, but in general it is the study of change. The study of operations and applications is algebra. The study of shape is geometry. The study of collections of data is statistics.

3. Find f'(x) (the derivative of f(x)) if f(x) = 7x^3 - 2x^2 + 4x - 5.

From Quiz Quest for Calculus

Answer: 21x^2 - 4x + 4

The derivative of a polynomial equation can be found by multiplying the exponent of the variable by the coefficient of each term, then by reducing the exponent of the variable by one.

4. What is the derivative of 'y = 3x squared'?

From Quiz Small Derivatives Quiz

Answer: 6x

The example illustrates the Power Rule for derivatives. Using the rule, you multiply the power of x by the constant before x and reduce the power of x by one to get the derivative.

5. What are the two main branches of calculus?

From Quiz Calculus Fun (or My Head Hurts!)

Answer: differential and integral

The Fundamental Theorem of Calculus relates the two branches by stating they are inverse operations of each other.

6. Take 'y = x^3 + 2x^2 + 4' to be the equation of a curve. If this equation is then differentiated with respect to 'x' it would become the derivative, 'dy/dx = 3x^2 + 4x'. What characteristic of the curve does this derivative represent?

From Quiz The Wonderful World of Differentiation

Answer: Gradient (slope)

As the line on the graph is a curve, the gradient will be different for different values of 'x'. If the graph showed a straight line then it would be possible to get an equation of the form 'y = mx + c' where 'm' is the constant gradient. With a curve it isn't quite as simple to obtain the gradient because you can't just take the value of 'm' from the equation as it doesn't exist! In order to obtain the gradient at a specific point on a curve you first have to differentiate the equation of the curve as is shown in the question. This new expression is known as the derivative and it is the derivative of 'y' in this case, with respect to 'x'. Then if you wish to find out the gradient at the point on the curve where 'x' equalled say 3, you just put 'x = 3' into the derivative. Thus for the curve 'y = x^3 + 2x^2 + 4' you can obtain the derivative 'dy/dx = 3x^2 + 4x' and the gradient at the point 'x = 3' is: dy/dx = 3(3)^2 + 4(3) dy/dx = 27 + 12 dy/dx = gradient = 39. If the gradient is 39 then you would expect to see a steep incline of the curve at that point.

7. Consider the parabola given by the equation y = x^2 + 4x - 5. At which point on the graph of this parabola is the slope of the tangent line equal to 10?

From Quiz The Tangent Line

Answer: (3,16)

The derivative of y = x^2 + 4x - 5 is given by dy/dx = 2x + 4. We want the value of x for which the slope of the tangent line equals 10, so we set the derivative equal to 10 and solve for x: 2x + 4 = 10 2x = 6 x = 3. Putting x = 3 into the equation of the parabola gives y = 16, so (3,16) is the desired point on the graph.

8. What is the derivative of 'f(y) = 2y to the power of one third'?

From Quiz Small Derivatives Quiz

Answer: two thirds y to the power of negative two thirds

The Power Rule as in the first question: multiply the power of x by the constant before x and reduce the power of x by one to get the derivative.

9. How many REAL roots does the polynomial x to the fifth +12x-7 have?

From Quiz Calculus Fundamentals

Answer: 1

This is where Rolle's theorem becomes really useful. We can tell the polynomial has at least one root since it goes to infinity as x does and negative infinity as x becomes very negative, so it has to hit 0 somewhere in between. If it had more than one root, then by Rolle's theorem between the two roots the derivative of the polynomial, 5x to the fourth+12, must be 0. But 5(x to the fourth)+12 is always positive, so there can't be two roots!

10. Who is widely considered to be the father of calculus (or the father of differential and integral calculus)?

From Quiz Calculus Fun (or My Head Hurts!)

Answer: Both

Today, they both share credit for developing differential and integral calculus. It has long been disputed as to who actually developed it first. Leibniz developed infinitesimal calculus independently and published his work first, but Newton began work on it before and took many years to publish his work. Leibniz focused more on systematic ways to solve problems and came up with appropriate symbols while Newton came up with rules, series, and functions in general.

11. The line y = 2x - 9 is tangent to the parabola y = x^2 + ax + b at the point (4,-1). What are the values of a and b?

From Quiz The Tangent Line

Answer: a = -6, b = 7

The derivative gives the slope of the tangent line. The derivative of y = x^2 + ax + b is given by: dy/dx = 2x + a The slope of the tangent line at the point (4,-1) is found by subsituting x = 4 into the derivative: We get 8 + a. This time we know that y = 2x - 9 is the tangent line, which has slope 2. Hence we have the equation 8 + a = 2, thus a = -6. To find b, note that the point (4,-1) lies on the graph of the parabola y = x^2 -6x + b, so substitute x = 4 and y = -1: -1 = 4^2 - 6*4 + b -1 = 16 - 24 + b -1 = -8 + b Thus b = 7. Note that this question is really backwards from the "usual" calculus questions!

12. Find dy/dx if y = e^a and a is constant.

From Quiz Quest for Calculus

Answer: 0

Since there is no x-dependence in the equation, y is equal to a constant. Therefore, the derivative equals zero.

13. What is the derivative of 'y = 2x squared - 3x + 4'?

From Quiz Small Derivatives Quiz

Answer: 4x - 3

This derivative can be found using the Sum Rule.

14. What is the integral from -Pi over 7 to Pi over 7 of Sin(x cubed)?

From Quiz Calculus Fundamentals

Answer: 0 & Zero & 0.0

Please say you did NOT try to actually find the antiderivative here! Sin(xcubed)=-Sin((-x)cubed), so the area below the axis to the left of 0 is the same as the area above the axis to the right, and vice versa, so everything cancels leaving 0. On a somewhat related note, if a teacher or professor ever gives you a really really ugly integral, try answering 0. If they ask you how you got it, smile and say 'by symmetry'. They might be impressed!

15. After you find the derivative of your curve, and find its value at a point of interest, the value may be positive, negative, or zero. What does it tell you about your curve if the derivative has a value of 0?

From Quiz You Derive Me Crazy!

Answer: The curve is horizontal at that point

A positive gradient means the curve is increasing in value as x increases, and the curve is going up; a negative value means it is going down. A vertical line does not have a gradient, while a horizontal one has a gradient of zero. This fact can be used to find the point where a curve has a maximum or minimum value - if it changes from having a positive gradient to having a negative one, then the point where the derivative has a value of 0 is a local maximum value; if it is changing from a negative value to a positive value, the point is a local minimum.

16. When differentiating a function, what does the answer mean? In other words, what does the derivative of a function represent?

From Quiz Calculus Fun (or My Head Hurts!)

Answer: both

The answer is both because there are two ways in defining the derivative of a function: physically and geometrically. The most common forms of differential notation are Leibniz notation, which is noted as dy/dx and the Legrange notation or prime notation, which is noted as f'(x). Both notations are defined as the derivative of y with respect to x.

17. If f(x) = x (ln x), find f'(x).

From Quiz Quest for Calculus

Answer: 1 + ln x

The rule for the derivative of a product is the first term times the derivative of the second term, plus the second term times the derivative of the first term.

18. What is the derivative of 'y = (2x + 3)squared'?

From Quiz Small Derivatives Quiz

Answer: 4 (2x + 3)

This is an example of how a Chain Rule is used. Firstly, you take the power of the bracket and multiply it by the bracket itself taking one power off the bracket. Then you take the derivative of the expression inside the bracket and multiply it by the number in front of the bracket.

19. What is the integral from 0 to Pi over 2 of (Sine of x divided by the sum of Sine of x and Cosine of x)?

From Quiz Calculus Fundamentals

Answer: Pi over 4

Symmetry strikes again! Call the integral I. If we replace x by (Pi over 2)-x, we see that the I is the integral of Cos(x) over ((Sin(x)+Cos(x)). Adding, 2I is the integral of (Cos(x)+Sin(x)) over (Sin(x)+Cos(x))= the integral of 1, which is Pi over 2, so I has value (Pi over 4), or about 0.78. No, I don't know what the antiderivative is, but who needs to find that anyway!

20. When integrating a function, what does the answer mean? In other words, what does the integral of a function represent?

From Quiz Calculus Fun (or My Head Hurts!)

Answer: the area under the graph of a funtion bounded by a closed interval [a,b]

Essentially, an integral is the anti-derivative. It is solved by taking the anti-derivative of the function with respect to x from the closed interval 'b' and subtracting the anti-derivative of the function with respect to x from the closed interval 'a'.

21. If the velocity of a particle at time t is represented by the equation v(t) = 8t + 2, find the position of the particle at time 3 if the position at time 0 is 0.

From Quiz Quest for Calculus

Answer: 42

The position function can be found by integrating the velocity funcion, which is 4t^2 + 2t + C. I said that the position at time 0 is 0, so 4(0) + 2(0) + C=0, which makes C = 0. Therefore, when t = 3, the position equals 4(9) + 2(3) = 36 + 6 = 42.

22. What is the derivative of 'f(x) = x(x + 1)'?

From Quiz Small Derivatives Quiz

Answer: 2x + 1

This can be solved using either the Product Rule or by multiplying x through the bracket and using the Sum Rule.

23. What does it mean to find the limit of a function?

From Quiz Calculus Fun (or My Head Hurts!)

Answer: the behavior of a function as it approaches a specific input

When determining the limit of a function, you are seeing what value (if any) the function is as it approaches a specific input. The limit may approach a constant, but it may also approach infinity or zero. Depending on the function, it may not even exist in either case.

24. At which points on the graph of the hyperbola x^2 - y^2 = 4 does the tangent line have slope equal to 5/4 ?

From Quiz The Tangent Line

Answer: (10/3, 8/3), (-10/3, -8/3)

First we need to find the derivative by implicit differentiation: x^2 - y^2 = 4 Differentiate everything with respect to x: 2x - 2y*dy/dx = 0 Solving for dy/dx gives dy/dx = x/y. Let (a,b) be a point on the graph where the tangent line has slope 5/4. Then a/b = 5/4 which means a = (5/4)b. (a,b) is a point on the hyperbola, so it satisfies the equation: a^2 - b^2 = 4 Now substitute a = (5/4)b and solve for b: (25/16)b^2 - b^2 = 4 (9/16)b^2 = 4 b^2 = 64/9 thus b = 8/3 or b = -8/3. Now use a = (5/4)b to get the value of a in each case. So we get the two points are (10/3, 8/3), (-10/3, -8/3). In case you're wondering about the other answers, note that all of the other answers really are points on the hyperbola! They were obtained by finding a rational parameterization of the hyperbola usiing the following trick: Start with the point (-2,0) on the hyperbola and draw a line with slope t through it. The other point of intersection has x-coordinate equal to x = 2(1 + t^2)/(1 - t^2).

25. An object is in freefall. It remains in freefall for 5 seconds before it hits the ground. How high was the object when it first started falling if its starting velocity was 0? (Acceleration due to gravity is -9.8 meters/second^2).

From Quiz Quest for Calculus

Answer: 122.5 meters

a(t) = -9.8 since acceleration due to gravity is -9.8 meters/second^2. The velocity function is found by integrating the acceleration function. v(t) = -9.8t + C. Since the velocity at time 0 is 0, C = 0. Integrating this gives the position function, x(t) = -4.9t^2 + C. When t = 5, the position is zero. So, -4.9(25) + C = 0 and C = 122.5, which is the original height of the object.

26. How many roots (real or complex and counting multiple roots the number of times they occur) will a polynomial of degree n have?

From Quiz Calculus Fundamentals

Answer: Exactly n

This is the fundamental theorem of Algebra. We don't know how many roots are real, but once we include complex roots, we can count them.

27. Consider 'x' to be a variable. What is the derivative of 12x^4 + 10x^3 - 5x^2 + 16 with respect to x?

From Quiz Calculus Fun (or My Head Hurts!)

Answer: 48x^3 + 30x^2 - 10x

The formula for finding a derivative of a single-variable equation is: f'(x) = (y*C)x^y-1, where C is a constant Therefore, the answer is obtained by applying the formula in this way: (4*12)x^4-1 + (3*10)x^3-1 - (5*2)x^2-1 + (0*16) = 48x^3 + 30x^2 - 10x

28. Suppose a function satisfies f(0)=1 and f(2)=-1. What can we say about the roots of the function between 0 and 2?

From Quiz Calculus Fundamentals

Answer: Nothing...we need more information

It pays to read the fine print on your theorems! The intermediate value theorem only applies for continuous functions, so the function could have any number of roots.

29. Consider 'x' to be a variable. What is the integral of 18x^2 - 10x + 3 with respect to x? Pick the most correct solution.

From Quiz Calculus Fun (or My Head Hurts!)

Answer: 6x^3 - 5x^2 + 3x + C

The formula for finding an integral of a single-variable equation is: F(x) = (Bx^y+1)/(y+1) + C, where B, C are constants Therefore, the answer is obtained by applying the formula in this way: (18x^2+1)/(2+1) - (10x^1+1)/(1+1) + (3x^0+1)/(0+1) + C = (18x^3)/3 - (10x^2)/2 + (3x)/1 + C = 6x^3 - 5x^2 + 3x + C

30. The cubic curve y = x^3 + ax^2 + bx + c passes through the point (1,3) and has tangent line y = x - 2 at the point (0,-2). What are the values of a, b, and c?

From Quiz The Tangent Line

Answer: a = 3, b = 1, c = -2

We know the point (1,3) lies on the curve, so putting x = 1 and y = 3 gives: 3 = 1 + a + b + c a + b + c = 2 Next, we know that (0,-2) lies on the curve, so putting x = 0 and y = -2 gives: -2 = c Substituting c = -2 above gives a + b - 2 = 2 or a + b = 4. We need one more equation. The derivative gives the slope of the tangent line: dy/dx = 3x^2 + 2ax + b Evaluating this at x = 0 gives b. But the slope of the tangent line y = x - 2 is 1. Thus b = 1. Substituting b = 1 into the equation a + b = 4 gives a = 3.

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Last Updated Feb 24 2024 5:46 AM
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