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Quiz: The Derivative Concept

Test your understanding of derivatives and rates of change with these review questions.

1. What does the derivative of a function represent?

  1. The average rate of change over an interval
  2. The instantaneous rate of change at a point
  3. The area under the curve
  4. The maximum value of the function
Show Answer

The correct answer is B. The derivative measures the instantaneous rate of change—how fast the function is changing at a specific moment. This is what distinguishes derivatives from average rates of change.

Concept Tested: Derivative Definition

2. Geometrically, the derivative at a point equals:

  1. The y-intercept of the graph
  2. The slope of the secant line
  3. The slope of the tangent line
  4. The area under the curve
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The correct answer is C. The derivative at a point equals the slope of the tangent line to the graph at that point. The tangent line touches the curve at exactly one point locally and shares its slope there.

Concept Tested: Tangent Line

3. What is the average rate of change of f(x) = x² from x = 1 to x = 4?

  1. 3
  2. 5
  3. 8
  4. 15
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The correct answer is B. Average rate of change = [f(4) − f(1)]/(4 − 1) = (16 − 1)/3 = 15/3 = 5. This is the slope of the secant line connecting (1, 1) and (4, 16).

Concept Tested: Average Rate of Change

4. What is the limit definition of the derivative?

  1. f'(x) = [f(b) − f(a)]/(b − a)
  2. f'(x) = lim(h→0) [f(x+h) − f(x)]/h
  3. f'(x) = f(x+1) − f(x)
  4. f'(x) = lim(x→∞) f(x)/x
Show Answer

The correct answer is B. The derivative is defined as f'(x) = lim(h→0) [f(x+h) − f(x)]/h, the limit of the difference quotient as the interval shrinks to zero. This captures the instantaneous rate of change.

Concept Tested: Limit Definition of Derivative

5. The expression [f(x+h) − f(x)]/h is called the:

  1. Derivative
  2. Secant slope
  3. Difference quotient
  4. All of the above
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The correct answer is C. The difference quotient [f(x+h) − f(x)]/h represents the slope of a secant line. When we take its limit as h→0, we get the derivative. It's the key expression in defining derivatives.

Concept Tested: Difference Quotient

6. Which notation represents the derivative of y with respect to x?

  1. dy/dx
  2. f'(x)
  3. Df
  4. All of the above
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The correct answer is D. All three notations represent the derivative: dy/dx (Leibniz notation), f'(x) (prime notation), and Df (operator notation). Different notations are useful in different contexts.

Concept Tested: Derivative Notation

7. If position is given by s(t), what does s'(t) represent?

  1. Acceleration
  2. Distance traveled
  3. Velocity
  4. Average speed
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The correct answer is C. The derivative of position with respect to time is velocity—the instantaneous rate of change of position. This is one of the most important physical interpretations of the derivative.

Concept Tested: Instantaneous Velocity

8. Using the limit definition, find f'(2) if f(x) = 3x + 1.

  1. 1
  2. 3
  3. 6
  4. 7
Show Answer

The correct answer is B. f'(x) = lim(h→0) [(3(x+h)+1) − (3x+1)]/h = lim(h→0) [3h]/h = 3. For linear functions, the derivative equals the slope, which is 3 everywhere.

Concept Tested: Derivative at a Point

9. The slope of the secant line through (a, f(a)) and (b, f(b)) represents:

  1. The derivative at a
  2. The derivative at b
  3. The average rate of change from a to b
  4. The instantaneous rate of change
Show Answer

The correct answer is C. The secant line connects two points on the curve, and its slope equals [f(b) − f(a)]/(b − a), the average rate of change over the interval. As the points get closer, the secant approaches the tangent.

Concept Tested: Secant Line

10. If f'(3) = −2, what does this tell us about f at x = 3?

  1. f(3) = −2
  2. The function is decreasing at x = 3
  3. The function has a minimum at x = 3
  4. The tangent line is horizontal at x = 3
Show Answer

The correct answer is B. A negative derivative means the function is decreasing at that point—the tangent line has a negative slope. The function value is falling as x increases past 3.

Concept Tested: Derivative Function