Calculus Semester 1 Review

Front

What are critical numbers in calculus?

Back

Critical numbers are values of x in the domain of a function where the derivative is either zero or undefined. They are important for finding local maxima and minima.

Front

How do you find the critical numbers of a function?

Back

To find critical numbers, take the derivative of the function, set it equal to zero, and solve for x. Also, check where the derivative is undefined.

Front

What is the derivative of f(x) = 2x^4 - 4x^2 + 1?

Back

f'(x) = 8x^3 - 8x.

Front

What are the increasing intervals of the function f(x) = 2x^4 - 4x^2 + 1?

Back

The increasing intervals are (0, 1) and (-1, 0).

Front

What is the power rule for differentiation?

Back

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

Front

Find the derivative of f(x) = (x^6 + 4)^5.

Back

f'(x) = 30x^5(x^6 + 4)^4.

Front

What is the product rule in calculus?

Back

The product rule states that if u(x) and v(x) are functions, then the derivative of their product is u'v + uv'.

Front

What is the quotient rule in calculus?

Back

The quotient rule states that if u(x) and v(x) are functions, then the derivative of their quotient is (u'v - uv')/v^2.

Front

What does it mean for a function to be increasing on an interval?

Back

A function is increasing on an interval if, for any two points x1 and x2 in that interval, if x1 < x2, then f(x1) < f(x2).

Front

What is the second derivative test?

Back

The second derivative test is used to determine the concavity of a function and to classify critical points as local maxima or minima.