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AP Calculus BC : Derivatives

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Example Questions

Example Question #345 : Ap Calculus Bc

$\begin{align*}&\text{Find any and all local maxima for the function}\\&8x + 2x^{2} - 4x^{3} - 5x^{4} + 2x^{5} + 8\end{align*}$

Possible Answers:

$-0.951,1.529,2.845\text{ = maxima.}$

$0.716,2.375\text{ = maxima.}$

$2.375\text{ = maxima.}$

$0.716\text{ = maxima.}$

Correct answer:

$0.716\text{ = maxima.}$

Explanation:

$\begin{align*}&\text{Local maxima of a function are relative peaks; points that have}\\&\text{no greater value on either immediate side of them. To find local}\\&\text{maxima, take the derivative of a function and find the roots}\\&\text{of this derivative: the points where it equals zero. If a point}\\&\text{is a local maxima, then derivative values immediately to the}\\&\text{left will be positive, and values to the immediate right will}\\&\text{be negative.}\\&\text{For our function }8x + 2x^{2} - 4x^{3} - 5x^{4} + 2x^{5} + 8\\&\text{The derivative is }4x - 12x^{2} - 20x^{3} + 10x^{4} + 8\\&\text{The roots of this function are }x=0.716,2.375\\&\text{Checking each, we find that:}\\&\text{The root at }x=0.716\text{ is a maximum.}\\&\text{The root at }x=2.375\text{ is a minimum.}\\&\\&0.716\text{ = maxima.}\end{align*}$

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Example Question #351 : Ap Calculus Bc

$\begin{align*}&\text{Determine any local maxima for the function}\\&f(x)=5 - 6x^{2} - 3x^{3} - 3x^{4} - 8x^{5} - 3x\end{align*}$

Possible Answers:

$-0.458,-0.346\text{ = maxima.}$

$-0.458\text{ = maxima.}$

$0.569\text{ = maxima.}$

$-0.346\text{ = maxima.}$

Correct answer:

$-0.346\text{ = maxima.}$

Explanation:

$\begin{align*}&\text{Local maxima of a function are relative peaks; points that have}\\&\text{no greater value on either immediate side of them. To find local}\\&\text{maxima, take the derivative of a function and find the roots}\\&\text{of this derivative: the points where it equals zero. If a point}\\&\text{is a local maxima, then derivative values immediately to the}\\&\text{left will be positive, and values to the immediate right will}\\&\text{be negative.}\\&\text{For our function }5 - 6x^{2} - 3x^{3} - 3x^{4} - 8x^{5} - 3x\\&\text{The derivative is }- 12x - 9x^{2} - 12x^{3} - 40x^{4} - 3\\&\text{The roots of this function are }x=-0.458,-0.346\\&\text{Checking each, we find that:}\\&\text{The root at }x=-0.458\text{ is a minimum.}\\&\text{The root at }x=-0.346\text{ is a maximum.}\\&\\&-0.346\text{ = maxima.}\end{align*}$

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Example Question #352 : Ap Calculus Bc

$\begin{align*}&\text{Find the local maxima for }\\&f(x)=2x^{2} - 3x + 4x^{3} - 3x^{4} + 3x^{5} - 2\end{align*}$

Possible Answers:

$-0.481\text{ = maxima.}$

$0.388\text{ = maxima.}$

$0.93\text{ = maxima.}$

$-0.481,0.388\text{ = maxima.}$

Correct answer:

$-0.481\text{ = maxima.}$

Explanation:

$\begin{align*}&\text{Local maxima of a function are relative peaks; points that have}\\&\text{no greater value on either immediate side of them. To find local}\\&\text{maxima, take the derivative of a function and find the roots}\\&\text{of this derivative: the points where it equals zero. If a point}\\&\text{is a local maxima, then derivative values immediately to the}\\&\text{left will be positive, and values to the immediate right will}\\&\text{be negative.}\\&\text{For our function }2x^{2} - 3x + 4x^{3} - 3x^{4} + 3x^{5} - 2\\&\text{The derivative is }4x + 12x^{2} - 12x^{3} + 15x^{4} - 3\\&\text{The roots of this function are }x=-0.481,0.388\\&\text{Checking each, we find that:}\\&\text{The root at }x=-0.481\text{ is a maximum.}\\&\text{The root at }x=0.388\text{ is a minimum.}\\&\\&-0.481\text{ = maxima.}\end{align*}$

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Example Question #353 : Ap Calculus Bc

$\begin{align*}&\text{Find any and all local maxima for the function}\\&5x^{2} - x + 6x^{3} - 4\end{align*}$

Possible Answers:

$-0.642,0.087\text{ = maxima.}$

$0.087\text{ = maxima.}$

$0.713\text{ = maxima.}$

$-0.642\text{ = maxima.}$

Correct answer:

$-0.642\text{ = maxima.}$

Explanation:

$\begin{align*}&\text{Local maxima of a function are relative peaks; points that have}\\&\text{no greater value on either immediate side of them. To find local}\\&\text{maxima, take the derivative of a function and find the roots}\\&\text{of this derivative: the points where it equals zero. If a point}\\&\text{is a local maxima, then derivative values immediately to the}\\&\text{left will be positive, and values to the immediate right will}\\&\text{be negative.}\\&\text{For our function }5x^{2} - x + 6x^{3} - 4\\&\text{The derivative is }10x + 18x^{2} - 1\\&\text{The roots of this function are }x=-0.642,0.087\\&\text{Checking each, we find that:}\\&\text{The root at }x=-0.642\text{ is a maximum.}\\&\text{The root at }x=0.087\text{ is a minimum.}\\&\\&-0.642\text{ = maxima.}\end{align*}$

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Example Question #354 : Ap Calculus Bc

$\begin{align*}&\text{Find the local maxima for} \\&f(x)=5x^{2} - 8x - 5x^{3} - 6x^{4} + x^{5} + 5\end{align*}$

Possible Answers:

$5.305\text{ = maxima.}$

$-1.618,0.618,6.662\text{ = maxima.}$

$-1.069\text{ = maxima.}$

$-1.069,5.305\text{ = maxima.}$

Correct answer:

$-1.069\text{ = maxima.}$

Explanation:

$\begin{align*}&\text{Local maxima of a function are relative peaks; points that have}\\&\text{no greater value on either immediate side of them. To find local}\\&\text{maxima, take the derivative of a function and find the roots}\\&\text{of this derivative: the points where it equals zero. If a point}\\&\text{is a local maxima, then derivative values immediately to the}\\&\text{left will be positive, and values to the immediate right will}\\&\text{be negative.}\\&\text{For our function }5x^{2} - 8x - 5x^{3} - 6x^{4} + x^{5} + 5\\&\text{The derivative is }10x - 15x^{2} - 24x^{3} + 5x^{4} - 8\\&\text{The roots of this function are }x=-1.069,5.305\\&\text{Checking each, we find that:}\\&\text{The root at }x=-1.069\text{ is a maximum.}\\&\text{The root at }x=5.305\text{ is a minimum.}\\&\\&-1.069\text{ = maxima.}\end{align*}$

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Example Question #355 : Ap Calculus Bc

$\begin{align*}&\text{Find any and all local maxima for the function}\\&3x^{3} - 6x^{2} - 4x - 8\end{align*}$

Possible Answers:

$1.609\text{ = maxima.}$

$2.812\text{ = maxima.}$

$-0.276\text{ = maxima.}$

$-0.276,1.609\text{ = maxima.}$

Correct answer:

$-0.276\text{ = maxima.}$

Explanation:

$\begin{align*}&\text{Local maxima of a function are relative peaks; points that have}\\&\text{no greater value on either immediate side of them. To find local}\\&\text{maxima, take the derivative of a function and find the roots}\\&\text{of this derivative: the points where it equals zero. If a point}\\&\text{is a local maxima, then derivative values immediately to the}\\&\text{left will be positive, and values to the immediate right will}\\&\text{be negative.}\\&\text{For our function }3x^{3} - 6x^{2} - 4x - 8\\&\text{The derivative is }9x^{2} - 12x - 4\\&\text{The roots of this function are }x=-0.276,1.609\\&\text{Checking each, we find that:}\\&\text{The root at }x=-0.276\text{ is a maximum.}\\&\text{The root at }x=1.609\text{ is a minimum.}\\&\\&-0.276\text{ = maxima.}\end{align*}$

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Example Question #356 : Ap Calculus Bc

$\begin{align*}&\text{Find the local maxima for} \\&f(x)=7x^{4} - 4x^{2} - 3x^{3} - x - 4\end{align*}$

Possible Answers:

$-0.25,0.76\text{ = maxima.}$

$-0.879,1.259\text{ = maxima.}$

$-0.25,-0.188,0.76\text{ = maxima.}$

$-0.188\text{ = maxima.}$

Correct answer:

$-0.188\text{ = maxima.}$

Explanation:

$\begin{align*}&\text{Local maxima of a function are relative peaks; points that have}\\&\text{no greater value on either immediate side of them. To find local}\\&\text{maxima, take the derivative of a function and find the roots}\\&\text{of this derivative: the points where it equals zero. If a point}\\&\text{is a local maxima, then derivative values immediately to the}\\&\text{left will be positive, and values to the immediate right will}\\&\text{be negative.}\\&\text{For our function }7x^{4} - 4x^{2} - 3x^{3} - x - 4\\&\text{The derivative is }28x^{3} - 9x^{2} - 8x - 1\\&\text{The roots of this function are }x=-0.25,-0.188,0.76\\&\text{Checking each, we find that:}\\&\text{The root at }x=-0.25\text{ is a minimum.}\\&\text{The root at }x=-0.188\text{ is a maximum.}\\&\text{The root at }x=0.76\text{ is a minimum.}\\&\\&-0.188\text{ = maxima.}\end{align*}$

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Example Question #357 : Ap Calculus Bc

$\begin{align*}&\text{Find the local maxima for} \\&f(x)=4x^{2} - 3x + 3x^{3} - 5x^{4} + 3x^{5} + 6\end{align*}$

Possible Answers:

$-1.054\text{ = maxima.}$

$-0.549,0.322\text{ = maxima.}$

$0.322\text{ = maxima.}$

$-0.549\text{ = maxima.}$

Correct answer:

$-0.549\text{ = maxima.}$

Explanation:

$\begin{align*}&\text{Local maxima of a function are relative peaks; points that have}\\&\text{no greater value on either immediate side of them. To find local}\\&\text{maxima, take the derivative of a function and find the roots}\\&\text{of this derivative: the points where it equals zero. If a point}\\&\text{is a local maxima, then derivative values immediately to the}\\&\text{left will be positive, and values to the immediate right will}\\&\text{be negative.}\\&\text{For our function }4x^{2} - 3x + 3x^{3} - 5x^{4} + 3x^{5} + 6\\&\text{The derivative is }8x + 9x^{2} - 20x^{3} + 15x^{4} - 3\\&\text{The roots of this function are }x=-0.549,0.322\\&\text{Checking each, we find that:}\\&\text{The root at }x=-0.549\text{ is a maximum.}\\&\text{The root at }x=0.322\text{ is a minimum.}\\&\\&-0.549\text{ = maxima.}\end{align*}$

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Example Question #86 : Derivatives

Determine the local minima of the following function:

$f(x)=\frac{x^4}{4}+\frac{5x^3}{3}-\frac{4x^3}{6}-5x^2$

Possible Answers:

The function has no local minima

x=-5, 2

x=-5

x=0

x=2

Correct answer:

x=-5, 2

Explanation:

To determine the local maxima of the function, we must determine the points at which the function's first derivative changes from positive to negative.

First, we must find the first derivative of the function:

f'(x)=x^3+5x^2-2x^2-10x

The derivative was found using the following rule:

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

Now, we must find the critical values, for which the first derivative is equal to zero:

x^3+5x^2-2x^2-10x=0

x^2(x+5)+2x(x+5)=0

(x^2-2x)(x+5)=0

c=-5, 0, 2

Using the critical values, we now create intervals over which to evaluate the sign of the first derivative:

$(-\infty, -5), (-5, 0), (0, 2), (2, \infty)$

Notice how at the bounds of the intervals, the first derivative is neither positive nor negative.

Evaluating the sign simply by plugging in any value on the given interval into the first derivative function, we find that on the first interval, the first derivative is negative, on the second interval, the first derivative is positive, on the third interval, the first derivative is negative, and on the fourth interval, the first derivative is positive. The values at which the first derivative changed from positive to negative were x=-5, 2 , thus a local maximum exists for those two values.

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Example Question #21 : Finding Maximums

Find the local maxima of the function if its first derivative is given by

f'=(x+4)(x+1)(x+2)

Possible Answers:

x=-4

x=-4, -1

x=-2

x=-4, -2, -1

x=2

Correct answer:

x=-2

Explanation:

To determine the local maxima of the function, we must determine the points at which the function's first derivative changes from positive to negative.

We are given the first derivative of the function; now we must find the critical values, at which the first derivative is equal to zero:

(x+4)(x+1)(x+2)=0

c=-4, -2, -1

Using the critical values, we now create intervals over which to evaluate the sign of the first derivative:

$(-\infty, -4), (-4, -2), (-2, -1), (-1, \infty)$

Notice how at the bounds of the intervals, the first derivative is neither positive nor negative.

Evaluating the sign simply by plugging in any value on the given interval into the first derivative function, we find that on the first interval, the first derivative is negative, on the second interval, the first derivative is positive, on the third interval, the first derivative is negative, and on the fourth interval, the first derivative is positive. The only value at which the first derivative changed from positive to negative was x=-2 , thus a local maximum exists here.

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AP Calculus BC : Derivatives

Study concepts, example questions & explanations for AP Calculus BC

All AP Calculus BC Resources

Example Questions

Example Question #345 : Ap Calculus Bc

Example Question #351 : Ap Calculus Bc

Example Question #352 : Ap Calculus Bc

Example Question #353 : Ap Calculus Bc

Example Question #354 : Ap Calculus Bc

Example Question #355 : Ap Calculus Bc

Example Question #356 : Ap Calculus Bc

Example Question #357 : Ap Calculus Bc

Example Question #86 : Derivatives

Example Question #21 : Finding Maximums

All AP Calculus BC Resources

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