AP Calculus BC Practice Quiz: Defining the Derivative of a Function and Using Derivative Notation

Correct Answer: B

The formal definition of the derivative of a function f is the limit of the difference quotient, which is given by the expression lim_{h -> 0} (f(x+h) - f(x)) / h. Option D represents the average rate of change, not the instantaneous rate of change.

Correct Answer: D

The standard notations for the derivative of y = f(x) include f'(x) (prime notation), y' (prime notation), and dy/dx (Leibniz notation). The notation f(x') is not a standard representation for the derivative.

Correct Answer: C

By definition, the derivative of a function at a point is the slope of the line tangent to the graph of the function at that point. It represents the instantaneous rate of change.

Correct Answer: A

The derivative is defined as lim_{h -> 0} (f(x+h) - f(x)) / h. For f(x) = 5x^2, we have f(x+h) = 5(x+h)^2. Substituting these into the definition gives the expression in option A.

Correct Answer: C

The given limit matches the form lim_{h -> 0} (f(a+h) - f(a)) / h. In this case, the function is f(x) = cos(x) and the point is a = π/2. Therefore, the limit represents f'(π/2) for f(x) = cos(x).

Correct Answer: B

The derivative of a function at a point, f'(a), is defined as the slope of the line tangent to the function's graph at x = a. Therefore, the slope of the tangent line at x = 3 is f'(3), which is -4.

Correct Answer: B

The expression matches the definition of the derivative, lim_{h -> 0} (f(x+h) - f(x)) / h, for the function f(x) = ln(3x), because f(x+h) would be ln(3(x+h)).

Correct Answer: B

Using a table of values to approximate the derivative is a numerical representation. An analytical representation would be a formula, a graphical one would be a graph of the derivative, and a verbal one would be a description in words.

Correct Answer: B

Using the point-slope form of a line, y - y1 = m(x - x1), with the point (1, 8) and slope m = -2, we get y - 8 = -2(x - 1). Simplifying gives y - 8 = -2x + 2, which leads to y = -2x + 10.

Correct Answer: B

The limit of the difference quotient is the fundamental definition of the derivative of a function, f'(x), which represents the instantaneous rate of change of the function.

Correct Answer: A

This limit is in the form lim_{x -> a} (f(x) - f(a)) / (x - a). Here, f(x) = sqrt(x), a = 4, and f(a) = f(4) = sqrt(4) = 2. Thus, the limit represents f'(4) for f(x) = sqrt(x).

Correct Answer: C

The derivative at a point is the slope of the tangent line at that point. For a straight line, the tangent line at any point is the line itself. Since the line has a constant slope of 5, its derivative must be the constant value 5 for all x.

Correct Answer: C

The derivative of a function at a point, g'(a), gives the instantaneous rate of change at x = a. Therefore, the statement that the instantaneous rate of change at x = -1 is 1/2 is written as g'(-1) = 1/2.

Correct Answer: A

Using the definition f'(x) = lim_{h -> 0} (f(x+h) - f(x)) / h, we substitute f(x) = 1/(x+3). This means f(x+h) = 1/((x+h)+3) = 1/(x+h+3). Plugging these into the formula gives the expression in option A. Option D is incorrect because it is missing the limit.

Correct Answer: B

The derivative of a function at a point, g'(-1), is the slope of the tangent line at that point. The equation of the tangent line is given in slope-intercept form, y = mx + b, where m is the slope. In this case, the slope is 5. Therefore, g'(-1) = 5.