AP Calculus AB Practice Quiz: Defining the Derivative of a Function and Using Derivative Notation
Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026
Test your understanding with short quizzes. This quiz has 15 questions to check your progress.
Question 1 of 15
All Questions (15)
A) lim(h->0) [f(x+h) - f(x)] / h
B) lim(x->a) [f(x+h) - f(x)] / h
C) [f(x+h) - f(x)] / h
D) lim(h->0) [f(x) - f(h)] / h
Correct Answer: A
The derivative of f is defined as the function whose value at x is the limit of the difference quotient, lim(h->0) [f(x+h) - f(x)] / h, provided the limit exists. [cite: 1637, 1638, 1640]
A) f'(x)
B) dy/dx
C) y'
D) f(x')
Correct Answer: D
Standard notations for the derivative of y = f(x) include f'(x), dy/dx, and y'. The notation f(x') is not a standard representation for the derivative. [cite: 1641, 1642]
A) The y-intercept of the function's graph.
B) The slope of the line tangent to the function's graph at that point.
C) The area under the curve at that point.
D) The value of the function at that point.
Correct Answer: B
A key interpretation of the derivative is that its value at a point is equal to the slope of the line tangent to the graph of the function at that same point. [cite: 1648]
A) f(x) = sin(x) at x = π/2
B) f(x) = cos(x) at x = π/2
C) f(x) = sin(x) at x = h
D) f(x) = sin(π/2) at x = 0
Correct Answer: A
This limit matches the form lim(h->0) [f(a+h) - f(a)] / h, which is the derivative of f at x=a. In this case, the function f(x) is sin(x) and the point a is π/2. [cite: 1637, 1638]
A) 3
B) 5
C) 1/5
D) Cannot be determined
Correct Answer: B
The notation f'(3) represents the value of the derivative of the function f at x=3. The derivative of a function at a point is the slope of the line tangent to the graph at that point. Therefore, the slope is 5. [cite: 1641, 1648]
A) y - 9 = 3(x - 9)
B) y - 3 = 6(x - 9)
C) y - 9 = 6(x - 3)
D) y - 9 = 2(x - 3)
Correct Answer: C
To find the equation of the tangent line, we need a point and a slope. The point is given as (3, 9). The slope is the value of the derivative at x=3, which is f'(3) = 2(3) = 6. Using the point-slope form y - y1 = m(x - x1), we get y - 9 = 6(x - 3). [cite: 1646, 1648]
A) f(x) = 3x^2
B) f(x) = x^3
C) f(x) = x
D) f(x) = h^3
Correct Answer: B
This expression matches the limit definition of the derivative, lim(h->0) [f(x+h) - f(x)] / h. By comparing the terms, we can see that f(x) = x^3. [cite: 1633, 1637]
A) Graphically (as the slope of a tangent line)
B) Numerically (as an approximation from a table of values)
C) Analytically (as a formula)
D) Hypothetically (as a theoretical possibility)
Correct Answer: D
The derivative can be represented graphically, numerically, analytically (as a formula or limit), and verbally. 'Hypothetically' is not a standard mathematical representation. [cite: 1644]
A) The value of the function g at x=2 is 4.
B) The line tangent to the graph of g at x=2 has a slope of 4.
C) The function g is a linear function with a slope of 4.
D) The y-intercept of the tangent line at x=2 is 4.
Correct Answer: B
The notation dy/dx represents the derivative of y with respect to x. Its value at a specific point is the slope of the tangent line to the function's graph at that point. [cite: 1641, 1648]
A) The tangent line to the graph of f at x.
B) The secant line through the points (x, f(x)) and (x+h, f(x+h)).
C) A horizontal line passing through f(x).
D) A vertical line passing through x.
Correct Answer: B
The difference quotient is the formula for the slope between two distinct points on a curve, which defines a secant line. The derivative is the limit of this slope as the distance between the points approaches zero. [cite: 1606, 1634]
A) y - 8 = 8(x - 2)
B) y - 2 = 12(x - 8)
C) y - 8 = 3(x - 2)
D) y - 8 = 12(x - 2)
Correct Answer: D
The problem asks for the equation of the tangent line. We are given the point (2, 8). We find the slope by evaluating the derivative at x=2: f'(2) = 3(2)^2 = 3(4) = 12. Using the point-slope form, the equation is y - 8 = 12(x - 2). [cite: 1646, 1648]
A) 1
B) 0
C) Undefined
D) Positive
Correct Answer: B
The derivative at a point is the slope of the tangent line at that point. A horizontal line has a slope of 0. Therefore, the derivative is 0 at any point where the tangent line is horizontal. [cite: 1648]
A) lim(h->0) [ (1/(x+h) - 1/x) / h ]
B) lim(h->0) [ (1/x - 1/(x+h)) / h ]
C) lim(h->0) [ (1/(x+h) + 1/x) / h ]
D) [ (1/(x+h) - 1/x) / h ]
Correct Answer: A
To represent the derivative of f(x) = 1/x, we substitute it into the definition f'(x) = lim(h->0) [f(x+h) - f(x)] / h. This gives lim(h->0) [ (1/(x+h) - 1/x) / h ]. Option B has the terms reversed, and Option D is missing the limit. [cite: 1633, 1637]
A) f(x) = x^2, a = 4
B) f(x) = x^2, a = 16
C) f(x) = 4x, a = 4
D) f(x) = 16, a = 4
Correct Answer: A
The expression matches the form lim(h->0) [f(a+h) - f(a)] / h. Here, (4+h)^2 corresponds to f(a+h), so f(x) = x^2 and a = 4. This is confirmed because f(a) = f(4) = 4^2 = 16, which matches the expression. [cite: 1637, 1638]
A) The graph has a value of -3.
B) The graph is decreasing.
C) The graph is increasing.
D) The graph has a y-intercept of -3.
Correct Answer: B
The notation y' represents the derivative, which is the slope of the tangent line. A negative slope (-3) indicates that the function is decreasing at that point. [cite: 1642, 1648]