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AP Calculus AB : Slope of a curve at a point

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Example Questions

Example Question #141 : Ap Calculus Ab

Functionplot

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

Possible Answers:

$\text{The derivative is negative.}$

$\text{The derivative is zero.}$

$\text{The derivative is positive.}$

Correct answer:

$\text{The derivative is negative.}$

Explanation:

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

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

Functionplot

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

Possible Answers:

$\text{The derivative is positive.}$

$\text{The derivative is zero.}$

$\text{The derivative is negative.}$

Correct answer:

$\text{The derivative is positive.}$

Explanation:

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

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Example Question #32 : Slope Of A Curve At A Point

Functionplot

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

Possible Answers:

$\text{The derivative is zero.}$

$\text{The derivative is positive.}$

$\text{The derivative is negative.}$

Correct answer:

$\text{The derivative is positive.}$

Explanation:

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

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Example Question #41 : Derivative At A Point

Functionplot

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

Possible Answers:

$\text{The derivative is negative.}$

$\text{The derivative is positive.}$

$\text{The derivative is zero.}$

Correct answer:

$\text{The derivative is positive.}$

Explanation:

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

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Example Question #32 : Slope Of A Curve At A Point

Functionplot

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

Possible Answers:

$\text{The derivative is zero.}$

$\text{The derivative is positive.}$

$\text{The derivative is negative.}$

Correct answer:

$\text{The derivative is positive.}$

Explanation:

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

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

Functionplot

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

Possible Answers:

$\text{The derivative is positive.}$

$\text{The derivative is zero}$

$\text{The derivative is negative.}$

Correct answer:

$\text{The derivative is zero}$

Explanation:

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

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

Functionplot

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

Possible Answers:

$\text{The derivative is negative.}$

$\text{The derivative is positive.}$

$\text{The derivative is zero.}$

Correct answer:

$\text{The derivative is positive.}$

Explanation:

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

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

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

f(x)=10x^3+1

Possible Answers:

271

181

180

270

Correct answer:

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

f'(x)=30x^2

and was found using the following rules:

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

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

f'(3)=30(3^2)=270

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Example Question #151 : Ap Calculus Ab

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

Possible Answers:

Correct answer:

Explanation:

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

y'=15x^4+9

We then evaluate ate the point x=1 ,

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

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Example Question #152 : Ap Calculus Ab

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

Possible Answers:

Correct answer:

Explanation:

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

Next, plug in x=1.

You should get k'(1)=1 .

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AP Calculus AB : Slope of a curve at a point

Study concepts, example questions & explanations for AP Calculus AB

All AP Calculus AB Resources

Example Questions

Example Question #141 : Ap Calculus Ab

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

Example Question #32 : Slope Of A Curve At A Point

Example Question #41 : Derivative At A Point

Example Question #32 : Slope Of A Curve At A Point

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

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

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

Example Question #151 : Ap Calculus Ab

Example Question #152 : Ap Calculus Ab

All AP Calculus AB Resources

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