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AP Calculus AB : AP Calculus AB

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3 Diagnostic Tests 164 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #31 : Slope Of A Curve At A Point

Functionplot

$\begin{align*}&\text{Determine whether or not the derivative of the function above at the point }\\&x=-1\\&\text{is positive, negative, or zero.}\end{align*}$ $\displaystyle \begin{align*}&\text{Determine whether or not the derivative of the function above at the point }\\&x=-1\\&\text{is positive, negative, or zero.}\end{align*}$

Possible Answers:

$\text{The derivative is positive.}$ $\displaystyle \text{The derivative is positive.}$

$\text{The derivative is negative.}$ $\displaystyle \text{The derivative is negative.}$

$\text{The derivative is zero.}$ $\displaystyle \text{The derivative is zero.}$

Correct answer:

$\text{The derivative is positive.}$ $\displaystyle \text{The derivative is positive.}$

Explanation:

$\begin{align*}&\text{The derivative of a function at a given point is synonymous}\\&\text{with the slope of a function at a given point. Therefore if}\\&\text{the function increases in value as x increases across a point,}\\&\text{the slope and derivative are positive. Conversely, if the function}\\&\text{decreases in value as x increases across a point, both slope}\\&\text{and derivative are negative. At points where the function is}\\&\text{flat, the slope and derivative are zero.}\\&\text{Studying the slope at }x=-1\text{, the function appears to be increasing.}\\&\text{The derivative is positive.}\end{align*}$ $\displaystyle \begin{align*}&\text{The derivative of a function at a given point is synonymous}\\&\text{with the slope of a function at a given point. Therefore if}\\&\text{the function increases in value as x increases across a point,}\\&\text{the slope and derivative are positive. Conversely, if the function}\\&\text{decreases in value as x increases across a point, both slope}\\&\text{and derivative are negative. At points where the function is}\\&\text{flat, the slope and derivative are zero.}\\&\text{Studying the slope at }x=-1\text{, the function appears to be increasing.}\\&\text{The derivative is positive.}\end{align*}$

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Example Question #151 : Ap Calculus Ab

Functionplot

$\begin{align*}&\text{Determine whether or not the derivative of the function above at the point }\\&x=7\\&\text{is positive, negative, or zero.}\end{align*}$ $\displaystyle \begin{align*}&\text{Determine whether or not the derivative of the function above at the point }\\&x=7\\&\text{is positive, negative, or zero.}\end{align*}$

Possible Answers:

$\text{The derivative is positive.}$ $\displaystyle \text{The derivative is positive.}$

$\text{The derivative is negative.}$ $\displaystyle \text{The derivative is negative.}$

$\text{The derivative is zero.}$ $\displaystyle \text{The derivative is zero.}$

Correct answer:

$\text{The derivative is positive.}$ $\displaystyle \text{The derivative is positive.}$

Explanation:

$\begin{align*}&\text{The derivative of a function at a given point is synonymous}\\&\text{with the slope of a function at a given point. Therefore if}\\&\text{the function increases in value as x increases across a point,}\\&\text{the slope and derivative are positive. Conversely, if the function}\\&\text{decreases in value as x increases across a point, both slope}\\&\text{and derivative are negative. At points where the function is}\\&\text{flat, the slope and derivative are zero.}\\&\text{Studying the slope at }x=7\text{, the function appears to be increasing.}\\&\text{The derivative is positive.}\end{align*}$ $\displaystyle \begin{align*}&\text{The derivative of a function at a given point is synonymous}\\&\text{with the slope of a function at a given point. Therefore if}\\&\text{the function increases in value as x increases across a point,}\\&\text{the slope and derivative are positive. Conversely, if the function}\\&\text{decreases in value as x increases across a point, both slope}\\&\text{and derivative are negative. At points where the function is}\\&\text{flat, the slope and derivative are zero.}\\&\text{Studying the slope at }x=7\text{, the function appears to be increasing.}\\&\text{The derivative is positive.}\end{align*}$

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Example Question #142 : Derivatives

Functionplot

$\begin{align*}&\text{Conclude if the derivative of the function above at the point }\\&x=3\\&\text{is positive, negative, or zero.}\end{align*}$ $\displaystyle \begin{align*}&\text{Conclude if the derivative of the function above at the point }\\&x=3\\&\text{is positive, negative, or zero.}\end{align*}$

Possible Answers:

$\text{The derivative is positive.}$ $\displaystyle \text{The derivative is positive.}$

$\text{The derivative is negative.}$ $\displaystyle \text{The derivative is negative.}$

$\text{The derivative is zero.}$ $\displaystyle \text{The derivative is zero.}$

Correct answer:

$\text{The derivative is positive.}$ $\displaystyle \text{The derivative is positive.}$

Explanation:

$\begin{align*}&\text{The derivative of a function at a given point is synonymous}\\&\text{with the slope of a function at a given point. Therefore if}\\&\text{the function increases in value as x increases across a point,}\\&\text{the slope and derivative are positive. Conversely, if the function}\\&\text{decreases in value as x increases across a point, both slope}\\&\text{and derivative are negative. At points where the function is}\\&\text{flat, the slope and derivative are zero.}\\&\text{Studying the slope at }x=3\text{, the function appears to be increasing.}\\&\text{The derivative is positive.}\end{align*}$ $\displaystyle \begin{align*}&\text{The derivative of a function at a given point is synonymous}\\&\text{with the slope of a function at a given point. Therefore if}\\&\text{the function increases in value as x increases across a point,}\\&\text{the slope and derivative are positive. Conversely, if the function}\\&\text{decreases in value as x increases across a point, both slope}\\&\text{and derivative are negative. At points where the function is}\\&\text{flat, the slope and derivative are zero.}\\&\text{Studying the slope at }x=3\text{, the function appears to be increasing.}\\&\text{The derivative is positive.}\end{align*}$

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Example Question #51 : Derivative At A Point

Functionplot

$\begin{align*}&\text{Conclude if the derivative of the function above at the point }\\&x=2\\&\text{is positive, negative, or zero.}\end{align*}$ $\displaystyle \begin{align*}&\text{Conclude if the derivative of the function above at the point }\\&x=2\\&\text{is positive, negative, or zero.}\end{align*}$

Possible Answers:

$\text{The derivative is zero.}$ $\displaystyle \text{The derivative is zero.}$

$\text{The derivative is negative.}$ $\displaystyle \text{The derivative is negative.}$

$\text{The derivative is positive.}$ $\displaystyle \text{The derivative is positive.}$

Correct answer:

$\text{The derivative is positive.}$ $\displaystyle \text{The derivative is positive.}$

Explanation:

$\begin{align*}&\text{The derivative of a function at a given point is synonymous}\\&\text{with the slope of a function at a given point. Therefore if}\\&\text{the function increases in value as x increases across a point,}\\&\text{the slope and derivative are positive. Conversely, if the function}\\&\text{decreases in value as x increases across a point, both slope}\\&\text{and derivative are negative. At points where the function is}\\&\text{flat, the slope and derivative are zero.}\\&\text{Studying the slope at }x=2\text{, the function appears to be increasing.}\\&\text{The derivative is positive.}\end{align*}$ $\displaystyle \begin{align*}&\text{The derivative of a function at a given point is synonymous}\\&\text{with the slope of a function at a given point. Therefore if}\\&\text{the function increases in value as x increases across a point,}\\&\text{the slope and derivative are positive. Conversely, if the function}\\&\text{decreases in value as x increases across a point, both slope}\\&\text{and derivative are negative. At points where the function is}\\&\text{flat, the slope and derivative are zero.}\\&\text{Studying the slope at }x=2\text{, the function appears to be increasing.}\\&\text{The derivative is positive.}\end{align*}$

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Example Question #33 : Slope Of A Curve At A Point

Functionplot

$\begin{align*}&\text{Conclude if the derivative of the function above at the point }\\&x=-0.2\\&\text{is positive, negative, or zero.}\end{align*}$ $\displaystyle \begin{align*}&\text{Conclude if the derivative of the function above at the point }\\&x=-0.2\\&\text{is positive, negative, or zero.}\end{align*}$

Possible Answers:

$\text{The derivative is positive.}$ $\displaystyle \text{The derivative is positive.}$

$\text{The derivative is zero}$ $\displaystyle \text{The derivative is zero}$

$\text{The derivative is negative.}$ $\displaystyle \text{The derivative is negative.}$

Correct answer:

$\text{The derivative is zero}$ $\displaystyle \text{The derivative is zero}$

Explanation:

$\begin{align*}&\text{The derivative of a function at a given point is synonymous}\\&\text{with the slope of a function at a given point. Therefore if}\\&\text{the function increases in value as x increases across a point,}\\&\text{the slope and derivative are positive. Conversely, if the function}\\&\text{decreases in value as x increases across a point, both slope}\\&\text{and derivative are negative. At points where the function is}\\&\text{flat, the slope and derivative are zero.}\\&\text{Studying the slope at }x=-0.2\text{, the function appears to be flat.}\\&\text{The derivative is zero}\end{align*}$ $\displaystyle \begin{align*}&\text{The derivative of a function at a given point is synonymous}\\&\text{with the slope of a function at a given point. Therefore if}\\&\text{the function increases in value as x increases across a point,}\\&\text{the slope and derivative are positive. Conversely, if the function}\\&\text{decreases in value as x increases across a point, both slope}\\&\text{and derivative are negative. At points where the function is}\\&\text{flat, the slope and derivative are zero.}\\&\text{Studying the slope at }x=-0.2\text{, the function appears to be flat.}\\&\text{The derivative is zero}\end{align*}$

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Example Question #151 : Ap Calculus Ab

Functionplot

$\begin{align*}&\text{Determine whether or not the derivative of the function above at the point }\\&x=-0.8\\&\text{is positive, negative, or zero.}\end{align*}$ $\displaystyle \begin{align*}&\text{Determine whether or not the derivative of the function above at the point }\\&x=-0.8\\&\text{is positive, negative, or zero.}\end{align*}$

Possible Answers:

$\text{The derivative is positive.}$ $\displaystyle \text{The derivative is positive.}$

$\text{The derivative is negative.}$ $\displaystyle \text{The derivative is negative.}$

$\text{The derivative is zero.}$ $\displaystyle \text{The derivative is zero.}$

Correct answer:

$\text{The derivative is positive.}$ $\displaystyle \text{The derivative is positive.}$

Explanation:

$\begin{align*}&\text{The derivative of a function at a given point is synonymous}\\&\text{with the slope of a function at a given point. Therefore if}\\&\text{the function increases in value as x increases across a point,}\\&\text{the slope and derivative are positive. Conversely, if the function}\\&\text{decreases in value as x increases across a point, both slope}\\&\text{and derivative are negative. At points where the function is}\\&\text{flat, the slope and derivative are zero.}\\&\text{Studying the slope at }x=-0.8\text{, the function appears to be increasing.}\\&\text{The derivative is positive.}\end{align*}$ $\displaystyle \begin{align*}&\text{The derivative of a function at a given point is synonymous}\\&\text{with the slope of a function at a given point. Therefore if}\\&\text{the function increases in value as x increases across a point,}\\&\text{the slope and derivative are positive. Conversely, if the function}\\&\text{decreases in value as x increases across a point, both slope}\\&\text{and derivative are negative. At points where the function is}\\&\text{flat, the slope and derivative are zero.}\\&\text{Studying the slope at }x=-0.8\text{, the function appears to be increasing.}\\&\text{The derivative is positive.}\end{align*}$

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Example Question #151 : Derivatives

Find the slope of the tangent line to the following function at x=3:

$\displaystyle f(x)=10x^3+1$

Possible Answers:

$\displaystyle 30$

180 $\displaystyle 180$

181 $\displaystyle 181$

270 $\displaystyle 270$

271 $\displaystyle 271$

Correct answer:

270 $\displaystyle 270$

Explanation:

The slope of the tangent line to a curve is given by the first derivative evaluated at the point of interest.

The first derivative of the function is equal to

$\displaystyle f'(x)=30x^2$

and was found using the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$ $\displaystyle \frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$

To find the slope of the tangent line at x=3, we simply evaluate the first derivative function at x=3:

$\displaystyle f'(3)=30(3^2)=270$

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Example Question #151 : Ap Calculus Ab

Find the slope of the curve $\displaystyle y=3x^5+9x+3$ at the point x=1 $\displaystyle x=1$

Possible Answers:

$\displaystyle 24$

$\displaystyle 27$

$\displaystyle 22$

$\displaystyle 20$

Correct answer:

$\displaystyle 24$

Explanation:

To find the slope of the curve at any point, we take its derivative

$\displaystyle y'=15x^4+9$

We then evaluate ate the point x=1 $\displaystyle x=1$,

$\displaystyle y'(1)=15(1)^4+9=24$

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Example Question #152 : Ap Calculus Ab

Find k'(1) if $k(x)=\frac{3x^2+4x-2}{10}$ $\displaystyle k(x)=\frac{3x^2+4x-2}{10}$.

Possible Answers:

$\displaystyle 8$

$\displaystyle .1$

$\displaystyle 1$

$\displaystyle 6$

$\displaystyle -1$

Correct answer:

$\displaystyle 1$

Explanation:

First, find the derivative. You should get $k'(x)=\frac{1}{10}(6x+4)$ $\displaystyle k'(x)=\frac{1}{10}(6x+4)$.

Next, plug in x=1.

You should get k'(1)=1 $\displaystyle k'(1)=1$.

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Example Question #41 : Slope Of A Curve At A Point

Find the slope of the curve at the specified point:

$\displaystyle y=2x^3+4x-10$ at the point x=1 $\displaystyle x=1$

Possible Answers:

$\displaystyle 8$

$\displaystyle 10$

$\displaystyle 12$

$\displaystyle 7$

Correct answer:

$\displaystyle 10$

Explanation:

To find the slope of the curve, we take the derivative of the function:

$y'=m=\frac{\mathrm{d} }{\mathrm{d} x}(2x^3+4x-10)=6x^2+4$ $\displaystyle y'=m=\frac{\mathrm{d} }{\mathrm{d} x}(2x^3+4x-10)=6x^2+4$

Evaluating the slope at the specified point, we get

$\displaystyle y'(1)=6(1^2)+4=10$

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AP Calculus AB : AP Calculus AB

Study concepts, example questions & explanations for AP Calculus AB

All AP Calculus AB Resources

Example Questions

Example Question #31 : Slope Of A Curve At A Point

Example Question #151 : Ap Calculus Ab

Example Question #142 : Derivatives

Example Question #51 : Derivative At A Point

Example Question #33 : Slope Of A Curve At A Point

Example Question #151 : Ap Calculus Ab

Example Question #151 : Derivatives

Example Question #151 : Ap Calculus Ab

Example Question #152 : Ap Calculus Ab

Example Question #41 : Slope Of A Curve At A Point

All AP Calculus AB Resources

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