Calculus Fun (or My Head Hurts!) Quiz

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.

Answer: differential and integral

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

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.

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.

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'.

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.

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

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

Answer: 7

To simplify the integral from 0 to 1, use this formula:

F(x) = F(b) - F(a), where a and b are the bounds (in this case, a=0 and b=1)

Therefore, the answer is obtained by applying the formula in this way:

[8x^3 + 3x^2 - 4x + C]0-->1

= [8(1) + 3(1) - 4(1) + C] - [8(0) + 3(0) - 4(0) + C]

= [8 + 3 - 4 + C] - [0 + 0 - 0 + C]

= [7 + C] - [C]

= 7

Answer: Simpson's Rule

A series is the sum of the terms in a sequence and can be definite or indefinite. Simpson's Rule is used to approximate definite integrals, not series.