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AP Calculus AB Unit 4-5: Applications of Derivatives & Curve Sketching — practice questions

Explore 57 free Unit 4-5: Applications of Derivatives & Curve Sketching questions in AP Calculus AB Mathematics. Open each question or exercise to read its content, reference answers and available explanation.

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Unit 4-5: Applications of Derivatives & Curve Sketching questions and exercises with answers

Showing questions 1–30 of 57.

A wire of length $L$ is bent to form a rectangle. The **maximum area** enclosed is:Read the answer and worked solution →Slope of tangent to y = f(x) at point (a, f(a)) is:Read the answer and worked solution →Slope of normal line to y = f(x) at x = a (if f'(a) ≠ 0):Read the answer and worked solution →At a stationary point of f, derivative f' equals:Read the answer and worked solution →Function f is INCREASING on interval (a, b) if:Read the answer and worked solution →Local maximum of f at x = c (if f'(c) = 0):Read the answer and worked solution →Mean Value Theorem says that for continuous f on [a, b] differentiable on (a, b), there exists c ∈ (a, b) with:Read the answer and worked solution →For y = x³ - 3x² + 2, find critical points:Read the answer and worked solution →Point of inflection of curve y = f(x) is where:Read the answer and worked solution →For y = sin x, find dy/dx at x = π/3:Read the answer and worked solution →Approximation by differentials: Δy ≈ f'(x) Δx. Using this, sqrt(25.5) is approximately:Read the answer and worked solution →Rate of change of area of a circle (A = πr²) with respect to radius:Read the answer and worked solution →If two functions f and g satisfy f'(x) = g'(x) for all x on an interval, then:Read the answer and worked solution →Rate of change of volume V = (4/3)πr³ of a sphere with radius:Read the answer and worked solution →Linear approximation of f(x) near x = a:Read the answer and worked solution →A spherical balloon's volume increases at 30 cm³/s. At r = 5 cm, how fast is the radius increasing?Read the answer and worked solution →Find equation of tangent to y = x² - 1 at point (2, 3):Read the answer and worked solution →Maximum area of a rectangle inscribed in a semicircle of radius R (base on diameter):Read the answer and worked solution →Use L'Hôpital's rule: lim_(x→0) (sin x - x)/x³ =Read the answer and worked solution →For f(x) = x² ln x on (0, ∞), find x where f has local minimum:Read the answer and worked solution →A 5 m ladder slides down a wall. Bottom moves outward at 1 m/s. Speed of top when bottom is 3 m from wall:Read the answer and worked solution →For y = e^x − x, find values where tangent is horizontal:Read the answer and worked solution →For f(x) = x³ - 3x² + 4, the function is increasing on intervals:Read the answer and worked solution →Find minimum value of f(x) = (sin x)² + (cos x)² + 2 (sin x)(cos x):Read the answer and worked solution →Volume of cylinder inscribed in sphere of radius R, expressed using h (height of cylinder):Read the answer and worked solution →Concavity test: f(x) = e^x is:Read the answer and worked solution →Find the rate at which the surface area of a balloon (sphere) is changing when radius is 10 cm and dr/dt = 0.1 cm/s:Read the answer and worked solution →If $y = f(x)$, then $\dfrac{dy}{dx}$ represents the:Read the answer and worked solution →The rate of change of the area of a circle ($A = \pi r^2$) with respect to its radius, when $r = 5$, is:Read the answer and worked solution →A function $f$ is strictly increasing on an interval if, throughout it:Read the answer and worked solution →

Worked practice questions

A wire of length $L$ is bent to form a rectangle. The **maximum area** enclosed is:

  1. $L^2/4$
  2. $L^2/8$
  3. $L^2/16$ (when shape becomes a square of side $L/4$)
  4. $L^2$
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Correct answer: C. $L^2/16$ (when shape becomes a square of side $L/4$)

Perimeter $= 2(x + y) = L$ ⇒ $y = L/2 - x$. Area $A = xy = x(L/2 - x)$. $dA/dx = L/2 - 2x = 0$ ⇒ $x = L/4$. Then $y = L/4$ (square!). Area = $(L/4)^2 = L^2/16$. (For fixed perimeter, square encloses max area; for fixed area, circle has min perimeter — isoperimetric inequality.)

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Slope of tangent to y = f(x) at point (a, f(a)) is:

  1. f(a)
  2. 1/f'(a)
  3. f''(a)
  4. f'(a) (derivative at the point)
Show answer and solution

Correct answer: D. f'(a) (derivative at the point)

Derivative f'(a) is the slope of the tangent line at x = a. **Equation of tangent:** y - f(a) = f'(a)(x - a).

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Slope of normal line to y = f(x) at x = a (if f'(a) ≠ 0):

  1. -1/f'(a) (perpendicular to tangent)
  2. f'(a)
  3. 1/f(a)
  4. -f'(a)
Show answer and solution

Correct answer: A. -1/f'(a) (perpendicular to tangent)

Normal ⊥ tangent. Slope of normal = -1/f'(a).

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More Mathematics topics

Unit 3B: Implicit Differentiation and Inverse FunctionsUnit 3A: Chain Rule and Composite FunctionsUnit 1A3: Function Composition and InversesUnit 1A1: Relations, Functions and GraphsUnit 1A2: Trigonometric Functions & Inverse TrigUnit 2: Differentiation and Fundamental Derivative RulesUnit 6A: Indefinite Integrals & Fundamental TheoremUnit 1B3: Asymptotic Behavior and Infinite Limits