Quiz Bowl You got to know it Ideas from Calculus (Mathematics)

1. This concept is defined as the value a function approaches as the input approaches another value, even if the function is not defined at that point. An example of this is the function 1x\frac{1}{x}x1​, where the limit as xxx approaches infinity is 0, even though the function never actually reaches 0. The formal definition of this concept was developed by Augustin-Louis Cauchy and is traditionally written with the Greek letters epsilon and delta. For 15 points, name this concept.

Answer: Limit

2. This concept refers to the ability of a function’s graph to be drawn without lifting the pen. Formally, a function is continuous at a point if it has a limit at that point and is equal to its limit there. It is common to refer to functions that are continuous at every point as continuous functions. Polynomials, exponential functions, and trigonometric functions like sine and cosine are continuous, while others, like tangent and secant, are not continuous at certain points. For 15 points, name this property of functions.

Answer: Continuity

3. This operation, commonly referred to as differentiation, takes a function and produces another function that describes the rate of change of the original function. At any given point, the value of this new function is equal to the slope of the tangent line to the original function’s graph at that point. It is denoted by either f′(x)f'(x)f′(x) or dfdx\frac{df}{dx}dxdf​. For 15 points, name this operation.

Answer: Derivative

4. A function is called this if its derivative can be evaluated. A function is continuous if it has no breaks, jumps, or holes, but this property is a stronger condition. While all differentiable functions are continuous, not all continuous functions are differentiable. An example of a function that is continuous but not differentiable is the absolute value function at x=0x = 0x=0, where the graph has a sharp corner. For 15 points, name this property of functions.

Answer: Differentiable

5. This operation is used to compute the signed area under a curve, between the curve and the x-axis, from one point to another. It is denoted by the integral symbol ∫\int∫ and typically performed between two specified points, such as from 0 to 5. For 15 points, name this operation.

Answer: Definite integration

6. This method approximates the area under a curve by dividing the region into narrow rectangles and adding up the areas of these rectangles. As the width of the rectangles approaches zero, the sum approaches the definite integral of the function. For 15 points, name this method.

Answer: Riemann sum

7. This fundamental theorem of calculus has two parts. The first part states that integrating a function gives an antiderivative of the function, and the second part states that the definite integral of a function from a to b can be calculated as F(b)−F(a)F(b) - F(a)F(b)−F(a), where FFF is an antiderivative of the function. For 15 points, name this theorem.

Answer: Fundamental theorem of calculus

8. This rule is used to differentiate a function that is the composition of two other functions. It states that the derivative of f(g(x))f(g(x))f(g(x)) is f′(g(x))⋅g′(x)f'(g(x)) \cdot g'(x)f′(g(x))⋅g′(x). In Leibniz notation, it is written as dfdx=dfdg⋅dgdx\frac{df}{dx} = \frac{df}{dg} \cdot \frac{dg}{dx}dxdf​=dgdf​⋅dxdg​. For 15 points, name this rule.

Answer: Chain rule

9. This rule is used to differentiate the product of two functions. It states that the derivative of f(x)g(x)f(x)g(x)f(x)g(x) is f′(x)g(x)+f(x)g′(x)f'(x)g(x) + f(x)g'(x)f′(x)g(x)+f(x)g′(x). By integrating both sides of this result, we obtain the method of integration by parts. For 15 points, name this rule.

Answer: Product rule

10. This series is used to approximate a differentiable function with an infinite sum of monomials. The coefficients of the series are found by taking derivatives of the function and applying factorials. When the series is centered at x=0x = 0x=0, it is called a Maclaurin series. For 15 points, name this series.

Answer: Taylor series

11. These equations relate a function to its derivatives, and can involve first, second, or higher derivatives. They are used to model a wide range of behaviors, including population dynamics, waves, and fluid flow. For 15 points, name these equations.

Answer: Differential equations