Skip to content

Quiz: Related Rates and Linear Approximation

Test your understanding of related rates and linear approximation with these review questions.

  1. Rates that have no connection
  2. Quantities that change with respect to a common variable (usually time)
  3. Only constant rates
  4. Rates that are always equal
Show Answer

The correct answer is B. Related rates problems involve multiple quantities that change with respect to a common variable, typically time. An equation relates the quantities, and implicit differentiation relates their rates.

Concept Tested: Related Rates

  1. Differentiate immediately
  2. Draw a diagram and identify all variables and given rates
  3. Solve for the unknown variable
  4. Plug in numbers
Show Answer

The correct answer is B. Always start by drawing a diagram, labeling variables, and identifying what rates are given and what rate you need to find. Then find an equation relating the quantities.

Concept Tested: Related Rates Setup

3. A ladder 10 ft long leans against a wall. If the bottom slides away at 2 ft/sec, how fast is the top sliding down when the bottom is 6 ft from the wall?

  1. 1.5 ft/sec
  2. 2 ft/sec
  3. 2.5 ft/sec
  4. 3 ft/sec
Show Answer

The correct answer is A. Use x² + y² = 100. Differentiate: 2x(dx/dt) + 2y(dy/dt) = 0. When x = 6, y = 8. So 2(6)(2) + 2(8)(dy/dt) = 0, giving dy/dt = −1.5 ft/sec (negative means sliding down).

Concept Tested: Ladder Problem

4. The linear approximation formula is:

  1. f(x) ≈ f(a) + f'(a)(x − a)
  2. f(x) ≈ f(a) + f''(a)(x − a)
  3. f(x) ≈ f'(a)(x − a)
  4. f(x) ≈ f(a)f'(x)
Show Answer

The correct answer is A. Linear approximation (tangent line approximation): L(x) = f(a) + f'(a)(x − a). This approximates f(x) near x = a using the tangent line at a.

Concept Tested: Linearization Formula

5. Using linear approximation, estimate √4.1 given that √4 = 2.

  1. 2.01
  2. 2.025
  3. 2.05
  4. 2.1
Show Answer

The correct answer is B. Let f(x) = √x, a = 4. f'(x) = 1/(2√x), f'(4) = 1/4. L(4.1) = 2 + (1/4)(0.1) = 2 + 0.025 = 2.025.

Concept Tested: Tangent Line Approximation

6. What is the differential dy if y = x³?

  1. dy = 3x²
  2. dy = 3x² dx
  3. dy = x³ dx
  4. dy = 3x dx
Show Answer

The correct answer is B. The differential dy = f'(x) dx. For y = x³, dy = 3x² dx. This represents the change in y along the tangent line for a small change dx in x.

Concept Tested: Differential

7. A spherical balloon is inflated so that its radius increases at 3 cm/sec. How fast is the volume increasing when r = 10 cm?

  1. 400π cm³/sec
  2. 1200π cm³/sec
  3. 4000π cm³/sec
  4. 1200 cm³/sec
Show Answer

The correct answer is B. V = (4/3)πr³. Differentiate: dV/dt = 4πr²(dr/dt). When r = 10, dr/dt = 3: dV/dt = 4π(100)(3) = 1200π cm³/sec.

Concept Tested: Balloon Problem

8. The error in linear approximation is generally smaller when:

  1. x is far from a
  2. x is close to a
  3. f'(a) is large
  4. f(a) is large
Show Answer

The correct answer is B. Linear approximation is most accurate when x is close to the center point a. The error grows as we move away from a, especially when the function has large curvature.

Concept Tested: Error in Approximation

9. A 5-foot tall person walks away from a 20-foot lamppost at 4 ft/sec. How fast is their shadow lengthening?

  1. 1 ft/sec
  2. 4/3 ft/sec
  3. 3 ft/sec
  4. 4 ft/sec
Show Answer

The correct answer is B. Using similar triangles: 20/(x+s) = 5/s, giving 20s = 5x + 5s, so s = x/3. Differentiate: ds/dt = (1/3)(dx/dt) = (1/3)(4) = 4/3 ft/sec.

Concept Tested: Shadow Problem

10. If dy = f'(x) dx, what does dy represent geometrically?

  1. The actual change in y
  2. The change in y along the tangent line
  3. The slope of the curve
  4. The area under the curve
Show Answer

The correct answer is B. The differential dy represents the change in y along the tangent line (linear approximation) for a change dx in x. It approximates but doesn't equal the actual change Δy.

Concept Tested: dy Notation