Calculus 1 Quiz: Estimating Derivatives
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Estimating DerivativesQuestion 1 of 3

Let f(x)f(x) be a function with f′′(x)<0f''(x)<0 for all xx. An estimate for f′(2)f'(2) is calculated using the formula E=f(2.1)−f(2)0.1E = \frac{f(2.1) - f(2)}{0.1}. Which of the following statements about the estimate EE must be true?

E<f′(2)E < f'(2)
E>f′(2)E > f'(2)
E=f′(2)E = f'(2)
The relationship between EE and f′(2)f'(2) cannot be determined from the given information.
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Calculus 1 Quiz

Calculus 1 Quiz: Estimating Derivatives

Practice Estimating Derivatives in Calculus 1 with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Estimating Derivatives, giving you a quick way to practice the rules, question types, and explanations that matter most for Calculus 1.

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Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

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Question 1

Let f(x)f(x) be a function with f′′(x)<0f''(x)<0 for all xx. An estimate for f′(2)f'(2) is calculated using the formula E=f(2.1)−f(2)0.1E = \frac{f(2.1) - f(2)}{0.1}. Which of the following statements about the estimate EE must be true?

  1. E<f′(2)E < f'(2) (correct answer)
  2. E>f′(2)E > f'(2)
  3. E=f′(2)E = f'(2)
  4. The relationship between EE and f′(2)f'(2) cannot be determined from the given information.
Explanation: The condition f′′(x)<0f''(x)<0 means that the function f(x)f(x) is concave down. For a concave down function, the graph lies below its tangent lines. Geometrically, this means the slope of the function is always decreasing. The estimate EE is the slope of the secant line connecting the points (2,f(2))(2, f(2)) and (2.1,f(2.1))(2.1, f(2.1)). Because the function's slope is decreasing over the interval [2,2.1][2, 2.1], the slope of the tangent line at the start of the interval, f′(2)f'(2), will be greater than the slope of the secant line over the interval. Therefore, E<f′(2)E < f'(2). The estimate is an underestimate.

Question 2

Consider the following three expressions used to estimate f′(2)f'(2) for a smooth function f(x)f(x):

I. E1=f(2.1)−f(2)0.1E_1 = \frac{f(2.1) - f(2)}{0.1}

II. E2=f(2.01)−f(1.99)0.02E_2 = \frac{f(2.01) - f(1.99)}{0.02}

III. E3=f(3)−f(1)2E_3 = \frac{f(3) - f(1)}{2}

Which of the following correctly orders these estimates from typically least accurate to typically most accurate?

  1. I, II, III
  2. III, II, I
  3. II, I, III
  4. III, I, II (correct answer)
Explanation: We need to compare the accuracy of the three estimates. Accuracy is determined by two factors: the type of difference quotient and the size of the interval (hh). E3E_3 is a symmetric difference quotient, but its interval is very wide ([1,3][1, 3], so h=1h=1). This will typically make it the least accurate estimate for the instantaneous rate of change at x=2x=2. E1E_1 is a forward difference quotient with a small interval (h=0.1h=0.1). This is better than E3E_3. E2E_2 is a symmetric difference quotient with a very small interval (h=0.01h=0.01). Symmetric difference quotients are generally more accurate than one-sided quotients, and smaller values of hh are generally more accurate than larger values. Thus, E2E_2 is the most accurate estimate. The order from least accurate to most accurate is therefore E3E_3, then E1E_1, then E2E_2.

Question 3

Let f(x)f(x) be a differentiable function for which the second derivative f′′(x)f''(x) exists. Which of the following expressions generally provides the most accurate estimate of f′(a)f'(a)?

  1. f(a+0.1)−f(a)0.1\frac{f(a+0.1) - f(a)}{0.1}
  2. f(a)−f(a−0.1)0.1\frac{f(a) - f(a-0.1)}{0.1}
  3. f(a+0.01)−f(a)0.01\frac{f(a+0.01) - f(a)}{0.01}
  4. f(a+0.01)−f(a−0.01)0.02\frac{f(a+0.01) - f(a-0.01)}{0.02} (correct answer)
Explanation: The accuracy of a finite difference approximation to a derivative depends on both the step size hh and the type of difference formula used. The expressions in A, B, and C are one-sided (forward or backward) difference quotients. The expression in D is a symmetric (or central) difference quotient. For a function that is sufficiently smooth, the error in a one-sided difference is proportional to the step size hh (i.e., error is O(h)O(h)). The error in a symmetric difference is proportional to the square of the step size (i.e., error is O(h2)O(h^2)). Because the error in the symmetric difference formula decreases much faster as h→0h \to 0, it is generally much more accurate than a one-sided difference, even if the step size in the one-sided formula is smaller. Therefore, the symmetric difference in choice D provides the best estimate.