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AP Calculus AB : Derivative at a point

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Example Question #41 : Derivative At A Point

Find the slope of the line tangent to the function

$g(x)=x^2\cos(x)$

at $x=\pi$

Possible Answers:

$2\pi$

$-2\pi$

$\pi$

Correct answer:

$-2\pi$

Explanation:

The slope of the line tangent to a function is nothing more than the first derivative, which is for this function equal to

$g'(x)=2x\cos(x)-x^2\sin(x)$

found using the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x} f(x)g(x)=f'(x)g(x)+f(x)g'(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x} \cos(x)=-\sin(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$

Evaluated at the given point, we find the slope of the tangent line equal to

$g'(\pi)=2\pi(\cos(\pi))-\pi^2(\sin(\pi))=-2\pi$

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Example Question #22 : Slope Of A Curve At A Point

Functionplot

$\begin{align*}&\text{Determine the sign of the derivative of the function above at the point }\\&x=2\end{align*}$

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 #141 : Ap Calculus Ab

Functionplot

$\begin{align*}&\text{Determine the sign of the derivative of the function above at the point }\\&x=4\end{align*}$

Possible Answers:

$\text{The derivative is positive.}$

$\text{The derivative is zero.}$

$\text{The derivative is negative.}$

Correct answer:

$\text{The derivative is negative.}$

Explanation:

$\begin{align*}&\text{The derivative of a function at a given point is synonymous}\\&\text{with the slope of a function at a given point. Therefore if}\\&\text{the function increases in value as x increases across a point,}\\&\text{the slope and derivative are positive. Conversely, if the function}\\&\text{decreases in value as x increases across a point, both slope}\\&\text{and derivative are negative. At points where the function is}\\&\text{flat, the slope and derivative are zero.}\\&\text{Studying the slope at }x=4\text{, the function appears to be decreasing.}\\&\text{The derivative is negative.}\end{align*}$

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Example Question #44 : Derivative At A Point

Functionplot

$\begin{align*}&\text{Determine the sign of the derivative of the function above at the point }\\&x=3\end{align*}$

Possible Answers:

$\text{The derivative is zero.}$

$\text{The derivative is negative.}$

$\text{The derivative is positive.}$

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 #45 : Derivative At A Point

Functionplot

$\begin{align*}&\text{Determine the sign of the derivative of the function above at the point }\\&x=-0.3\end{align*}$

Possible Answers:

$\text{The derivative is zero}$

$\text{The derivative is negative.}$

$\text{The derivative is positive.}$

Correct answer:

$\text{The derivative is zero}$

Explanation:

$\begin{align*}&\text{The derivative of a function at a given point is synonymous}\\&\text{with the slope of a function at a given point. Therefore if}\\&\text{the function increases in value as x increases across a point,}\\&\text{the slope and derivative are positive. Conversely, if the function}\\&\text{decreases in value as x increases across a point, both slope}\\&\text{and derivative are negative. At points where the function is}\\&\text{flat, the slope and derivative are zero.}\\&\text{Studying the slope at }x=-0.3\text{, the function appears to be flat.}\\&\text{The derivative is zero}\end{align*}$

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Example Question #46 : Derivative At A Point

Functionplot

$\begin{align*}&\text{Determine the sign of the derivative of the function above at the point }\\&x=0.2\end{align*}$

Possible Answers:

$\text{The derivative is zero}$

$\text{The derivative is negative.}$

$\text{The derivative is positive.}$

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 #47 : Derivative 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:

$\text{The derivative is zero.}$

$\text{The derivative is negative.}$

$\text{The derivative is positive.}$

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 #48 : Derivative 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:

$\text{The derivative is negative.}$

$\text{The derivative is positive.}$

$\text{The derivative is zero.}$

Correct answer:

$\text{The derivative is positive.}$

Explanation:

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

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Example Question #49 : Derivative 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:

$\text{The derivative is negative.}$

$\text{The derivative is zero.}$

$\text{The derivative is positive.}$

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 #141 : Derivatives

Functionplot

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

Possible Answers:

$\text{The derivative is positive.}$

$\text{The derivative is negative.}$

$\text{The derivative is zero.}$

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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AP Calculus AB : Derivative at a point

Study concepts, example questions & explanations for AP Calculus AB

All AP Calculus AB Resources

Example Questions

Example Question #41 : Derivative At A Point

Example Question #22 : Slope Of A Curve At A Point

Example Question #141 : Ap Calculus Ab

Example Question #44 : Derivative At A Point

Example Question #45 : Derivative At A Point

Example Question #46 : Derivative At A Point

Example Question #47 : Derivative At A Point

Example Question #48 : Derivative At A Point

Example Question #49 : Derivative At A Point

Example Question #141 : Derivatives

All AP Calculus AB Resources

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