Interim Exam #2 Review

Front

If g(2) = -1 and g'(2) = 4, find the equation of the tangent line when x = 2.

Back

The equation of the tangent line is given by the point-slope form: y - g(2) = g'(2)(x - 2). Thus, y + 1 = 4(x - 2).

Front

Back

Using the chain rule, the derivative is -72x^{-7} ⋅ cos(4x^{-6}) ⋅ sin^2(4x^{-6}).

Front

Back

Front

What can be inferred about a twice-differentiable function f at x = c if f is increasing, has an inflection point, and a y-intercept of -5?

Back

At x = c, it holds that f(c) < f''(c) < f'(c).

Front

Back

Front

Define the point-slope form of a linear equation.

Back

The point-slope form is given by y - y_1 = m(x - x_1), where m is the slope and (x_1, y_1) is a point on the line.

Front

What is the chain rule in differentiation?

Back

The chain rule states that if a function y = f(g(x)), then the derivative is dy/dx = f'(g(x)) * g'(x).

Front

Explain what an inflection point is.

Back

An inflection point is a point on a curve where the curvature changes sign, indicating a change in the concavity of the function.

Front

What does it mean for a function to be increasing at a point?

Back

A function is increasing at a point if its derivative at that point is positive, indicating that the function's value is rising.

Front

State the product rule for differentiation.

Back

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