Derivative Review

Front

What does it mean for a function to be continuous but not differentiable?

Back

A function is continuous at a point if there are no breaks, jumps, or holes in the graph at that point. However, it may not be differentiable if it has a sharp corner or cusp at that point.

Front

How do you find the equation of the tangent line to a function at a given point?

Back

To find the equation of the tangent line, first calculate the derivative of the function to find the slope at the given point. Then use the point-slope form of the line: y - y1 = m(x - x1), where (x1, y1) is the point on the curve.

Front

What is implicit differentiation?

Back

Implicit differentiation is a technique used to find the derivative of a function defined implicitly by an equation involving both x and y, without solving for y explicitly.

Front

What does the derivative of a function represent?

Back

The derivative represents the slope of the tangent line to the graph of the function at a given point, indicating the rate of change of the function with respect to its variable.

Front

Find the derivative of y = 2x^2(3x - 4).

Back

The derivative is 18x^2 - 16x.

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), where n is a real number.

Front

What is the product rule for differentiation?

Back

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

Front

What is the quotient rule for differentiation?

Back

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

Front

What is a critical point?

Back

A critical point of a function occurs where its derivative is zero or undefined, indicating potential local maxima, minima, or points of inflection.

Front

What is the second derivative test?

Back

The second derivative test is used to determine the concavity of a function at a critical point. If f''(x) > 0, the function is concave up (local minimum); if f''(x) < 0, it is concave down (local maximum).