alleryA function $f$ is strictly decreasing on an interval if, throughout it:
- $f'(x) < 0$
- $f'(x) = 0$
- $f'(x) > 0$
- $f(x) < 0$
Show answer and solution
Correct answer: A. $f'(x) < 0$
A function is strictly decreasing where its derivative is negative.
AP Calculus AB Unit 4-5: Applications of Derivatives & Curve Sketching — practice questions — page 2
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.
Practise Unit 4-5: Applications of Derivatives & Curve Sketching with Quill →Unit 4-5: Applications of Derivatives & Curve Sketching questions and exercises with answers
Showing questions 31–57 of 57.
A function $f$ is strictly decreasing on an interval if, throughout it:Read the answer and worked solution →The slope of the tangent to the curve $y = f(x)$ at the point $x = a$ is:Read the answer and worked solution →The slope of the normal to the curve $y = f(x)$ at $x = a$ is:Read the answer and worked solution →At a critical point (a candidate for a local maximum or minimum) of a differentiable function $f$:Read the answer and worked solution →The function $f(x) = x^2$ is strictly increasing for:Read the answer and worked solution →The rate of change of the volume of a sphere $V = \tfrac{4}{3}\pi r^3$ with respect to its radius $r$ is:Read the answer and worked solution →For $f(x) = x^2 - 4x + 3$, the critical point occurs at:Read the answer and worked solution →The slope of the tangent to $y = x^2$ at the point $(1,1)$ is:Read the answer and worked solution →By the second derivative test, if $f'(c) = 0$ and $f''(c) > 0$, then $c$ is a point of:Read the answer and worked solution →By the second derivative test, at a point of local maximum the value of $f''(x)$ is:Read the answer and worked solution →If $y = a \ln x + b x^2 + x$ has extreme values at $x = 1$ and $x = 2$, then $(a, b)$ is:Read the answer and worked solution →Find the **maximum value** of $f(x) = \sin x + \cos x$ for $x \in [0, 2\pi]$.Read the answer and worked solution →The volume of a sphere is increasing at $8\pi$ cm³/s. The rate of change of surface area when the radius is 2 cm is:Read the answer and worked solution →The slope of the tangent to $y = x^3 - 3x + 1$ at $x = 2$ is:Read the answer and worked solution →At a **local maximum** of a differentiable function $f$:Read the answer and worked solution →The **slope of the tangent** to the curve $y = f(x)$ at point $(a, f(a))$ is:Read the answer and worked solution →A function $f$ is **strictly increasing** on an interval if, on that interval:Read the answer and worked solution →Water is poured at $36\,\text{m}^3/\text{s}$ into a cylindrical vessel of base radius 3 m. The rate at which water level rises is:Read the answer and worked solution →If $f(x) = x^2 - 4x + 3$, the **interval** on which $f$ is decreasing is:Read the answer and worked solution →Find the approximate value of $\sqrt{25.5}$ using differentials (given $\sqrt{25} = 5$):Read the answer and worked solution →$f(x) = 2x^3 - 3x^2 - 12x + 5$. The **local maximum value** occurs at:Read the answer and worked solution →The function $f(x) = x^3 - 3x + 1$ has critical points at:Read the answer and worked solution →A rectangle is inscribed in a semicircle of radius $R$ with base on the diameter. The **maximum area** is:Read the answer and worked solution →Find the **shortest distance** from origin to the curve $xy = 4$ (in the first quadrant):Read the answer and worked solution →The function $f(x) = \dfrac{x}{1 + x^2}$ attains its maximum value on $[0, \infty)$ at:Read the answer and worked solution →If $f(x) = x^x$ for $x > 0$, then $f$ has a minimum at $x = $:Read the answer and worked solution →If the tangent to the curve $y = e^{2x}$ at point $(a, e^{2a})$ passes through origin, then $a$ equals:Read the answer and worked solution →
Worked practice questions
The slope of the tangent to the curve $y = f(x)$ at the point $x = a$ is:
- $\dfrac{1}{f'(a)}$
- $f(a)$
- $-\dfrac{1}{f'(a)}$
- $f'(a)$
Show answer and solution
Correct answer: D. $f'(a)$
The derivative f'(a) is the slope of the tangent at x = a.
The slope of the normal to the curve $y = f(x)$ at $x = a$ is:
- $f(a)$
- $\dfrac{1}{f(a)}$
- $f'(a)$
- $-\dfrac{1}{f'(a)}$
Show answer and solution
Correct answer: D. $-\dfrac{1}{f'(a)}$
The normal is perpendicular to the tangent, so its slope is −1/f'(a).
How to practise
- Choose a Unit 4-5: Applications of Derivatives & Curve Sketching question and try to work out your answer.
- Open the question to check the correct answer and read its worked solution.
- Continue in the app for bite-sized lessons, explanations and review with Quill. You can start without creating an account.
Put it into practice
Continue with Unit 4-5: Applications of Derivatives & Curve Sketching practice for AP Calculus AB. Read explanations as you go. No signup form needed to start.
Practise Unit 4-5: Applications of Derivatives & Curve Sketching →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