Unit 3 Calculus Test

Front

What is the definition of a derivative in calculus?

Back

The derivative of a function at a point is the limit of the average rate of change of the function as the interval approaches zero. It represents the slope of the tangent line to the function's graph at that point.

Front

Explain the concept of limits in calculus.

Back

A limit is a value that a function approaches as the input approaches some value. Limits are fundamental in defining derivatives and integrals.

Front

What is the power rule for differentiation?

Back

The power rule states that if f(x) = x^n, then the derivative f'(x) = n*x^(n-1).

Front

Define the term 'critical point' in the context of calculus.

Back

A critical point of a function is a point where the derivative is either zero or undefined. Critical points are used to find local maxima and minima.

Front

What is the difference between a local maximum and a local minimum?

Back

A local maximum is a point where the function value is higher than all nearby points, while a local minimum is a point where the function value is lower than all nearby points.

Front

State the Fundamental Theorem of Calculus.

Back

The Fundamental Theorem of Calculus links the concept of differentiation and integration, stating that if F is an antiderivative of f on an interval [a, b], then ∫_a^b f(x) dx = F(b) - F(a).

Front

What is the chain rule in differentiation?

Back

The chain rule is a formula for computing the derivative of the composition of two or more functions. If y = f(g(x)), then dy/dx = f'(g(x)) * g'(x).

Front

Explain the concept of concavity in relation to the second derivative.

Back

Concavity refers to the direction of the curvature of a function's graph. If the second derivative is positive, the graph is concave up; if negative, it is concave down.

Front

What is an inflection point?

Back

An inflection point is a point on the graph of a function where the concavity changes, which occurs when the second derivative is zero or undefined.

Front

Define the term 'asymptote' in calculus.

Back

An asymptote is a line that a graph approaches but never touches. There are vertical, horizontal, and oblique asymptotes.