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AP Calculus AB Unit 2: Differentiation and Fundamental Derivative Rules — practice questions — page 2

Explore 93 free Unit 2: Differentiation and Fundamental Derivative Rules 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 2: Differentiation and Fundamental Derivative Rules questions and exercises with answers

Showing questions 31–60 of 93.

If $(2,-3,6)$ are direction ratios of a line, then a set of direction cosines isRead the answer and worked solution →If $(l,m,n)$ are the direction cosines of a line, then which relation must hold?Read the answer and worked solution →If $vec{a}$, $vec{b}$ and $vec{c}$ are coplanar, then the scalar triple product satisfiesRead the answer and worked solution →The projection of vector $vec{a}$ on vector $vec{b}$ isRead the answer and worked solution →For $n$ even, the maximum value among the binomial coefficients ${}^nC_r$ occurs atRead the answer and worked solution →The number of positive integral solutions of $x_1+x_2+\cdots+x_r=n$ isRead the answer and worked solution →Three numbers are in harmonic progression if and only if their reciprocals are inRead the answer and worked solution →If $S_n=an^2+bn+c$, then the $n$th term $t_n$ of the sequence isRead the answer and worked solution →If $z=x+iy$, what is the modulus $|z|$? ![](https://qallery.app/diagrams/v2_337fce5574f0/img-003.jpeg)Read the answer and worked solution →Air is pumped into a balloon and both its radius and volume grow together. Linking the two rates of change is a problem in:Read the answer and worked solution →A student replaces a curve near one point with its tangent line to estimate values nearby. That estimate is the:Read the answer and worked solution →A point where the derivative is zero or undefined is called a:Read the answer and worked solution →If a function has a local extremum at a point, that point must be a:Read the answer and worked solution →The converse fails: a function with a critical point is not guaranteed to have there a:Read the answer and worked solution →One theorem guarantees a continuous function on a closed, bounded interval attains both a highest and a lowest value. It is the:Read the answer and worked solution →A special case of the Mean Value Theorem, applying when the endpoints have equal function values, is:Read the answer and worked solution →To classify a critical point by looking at how the derivative's sign changes across it, a student uses the:Read the answer and worked solution →A function is defined to be concave up on an interval when its derivative is:Read the answer and worked solution →By the same logic a twice-differentiable function is concave down when its derivative is:Read the answer and worked solution →Checking concavity through the sign of the second derivative is known as the:Read the answer and worked solution →A point where the graph switches from concave up to concave down is a point of:Read the answer and worked solution →Using the sign of the second derivative at a critical point to classify it is the:Read the answer and worked solution →A farmer wants the largest area from a fixed length of fencing. Setting up and solving that kind of problem is called:Read the answer and worked solution →A limit produces the form zero over zero, so its behaviour cannot be read off directly. That form is called:Read the answer and worked solution →The rule that resolves such a limit by differentiating numerator and denominator separately is:Read the answer and worked solution →An iterative technique for finding zeroes, built on tangent line approximations, is:Read the answer and worked solution →A function whose derivative is the given function is called its:Read the answer and worked solution →Knowing an object's velocity, a student recovers its position by finding an:Read the answer and worked solution →The notation used for a family of antiderivatives, complete with its constant, is the:Read the answer and worked solution →A tank is draining and a student is asked how fast the depth falls when the volume is changing at a known rate. That is again a problem of:Read the answer and worked solution →

Worked practice questions

If $(2,-3,6)$ are direction ratios of a line, then a set of direction cosines is

  1. $\left(\dfrac{2}{7},-\dfrac{3}{7},\dfrac{6}{7}\right)$
  2. $\left(\dfrac{2}{\sqrt{49}},-\dfrac{3}{\sqrt{49}},\dfrac{6}{\sqrt{49}}\right)$
  3. $\left(\dfrac{2}{\sqrt{13}},-\dfrac{3}{\sqrt{13}},\dfrac{6}{\sqrt{13}}\right)$
  4. $\left(\dfrac{2}{\sqrt{14}},-\dfrac{3}{\sqrt{14}},\dfrac{6}{\sqrt{14}}\right)$
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Correct answer: A. $\left(\dfrac{2}{7},-\dfrac{3}{7},\dfrac{6}{7}\right)$

Direction cosines are obtained by dividing the direction ratios by their magnitude. For $(2,-3,6)$, the magnitude is $\sqrt{2^2+(-3)^2+6^2}=\sqrt{49}=7$. Hence the direction cosines are $\left(\dfrac{2}{7},-\dfrac{3}{7},\dfrac{6}{7}\right)$. Option A and B are numerically the same, but A is the simplified standard form since exactly one option must be correct, A is the intended answer.

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If $(l,m,n)$ are the direction cosines of a line, then which relation must hold?

  1. $l+m+n=1$
  2. $l^2+m^2+n^2=1$
  3. $l^2-m^2+n^2=1$
  4. $lm+mn+nl=1$
Show answer and solution

Correct answer: B. $l^2+m^2+n^2=1$

Direction cosines are the cosines of the angles the line makes with the coordinate axes. **Therefore they satisfy the standard identity $l^2+m^2+n^2=1$.**

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If $vec{a}$, $vec{b}$ and $vec{c}$ are coplanar, then the scalar triple product satisfies

  1. $\vec{a}\cdot(\vec{b}\times\vec{c})=1$
  2. $\vec{a}\cdot(\vec{b}\times\vec{c})=-1$
  3. $\vec{a}\cdot(\vec{b}\times\vec{c})=0$
  4. $\vec{a}\cdot(\vec{b}\times\vec{c})=|\vec{a}||\vec{b}||\vec{c}|$
Show answer and solution

Correct answer: C. $\vec{a}\cdot(\vec{b}\times\vec{c})=0$

The scalar triple product gives the signed volume of the parallelepiped formed by the three vectors. If the vectors are coplanar, that volume is zero, so $\vec{a}\cdot(\vec{b}\times\vec{c})=0$.

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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 6A: Indefinite Integrals & Fundamental TheoremUnit 1B3: Asymptotic Behavior and Infinite LimitsUnit 1B1: Limit Concepts and Graphical Limits