Calculus Trivia

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.

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.

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.

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.

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.

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.

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.

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.

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!

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.

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!

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.

From Quiz Small Derivatives Quiz

Answer: 4x - 3

This derivative can be found using the Sum Rule.

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!

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.

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.

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.

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.

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!

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'.

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.

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.

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.

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).

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.

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.

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

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.

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

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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