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微积分AB:确定当地的线性和线性ization

Study concepts, example questions & explanations for Calculus AB

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Example Question #1 :Determine Local Linearity And Linearization

Differentiate,

$y = \sin(\sin(\sin\(3x)))$

Possible Answers:

$\frac{dy}{dx}=3\cos(\sin(\sin(x)))\times \sin(\sin(x))\times \cos(3x)$

$\frac{dy}{dx}=3\cos(\cos(\sin(3x)))\times \sin(\sin(3x))\times \sin(3x)$

$\frac{dy}{dx}=3\cos(\sin(\sin(3x)))\times \cos(\sin(3x))$

$\frac{dy}{dx}=3\cos(\sin(\sin(3x)))\times \cos(\sin(3x))\times \cos(3x)$

$\frac{dy}{dx}=3\cos(\cos(\sin(3x)))\times \cos(\sin(3x))$

Correct answer:

$\frac{dy}{dx}=3\cos(\sin(\sin(3x)))\times \cos(\sin(3x))\times \cos(3x)$

Explanation:

Differentiate,

$y = \sin(\sin(\sin\(3x)))$

Strategy

This one at first glance appears difficult even if we recognize that the chain rule is needed; we have a function within a function within a function within a function. To avoid making mistakes, it's best to start by defining variables to make the calculation easier to follow.

Let's start with the outermost function, we will writeas a function ofby setting,

$u = \sin(\sin(3x))$

______________________________________________________

$y = \sin(\underset{u}{ \underbrace{\sin(\sin(3x))}}) = \sin(u)$

$y = \sin(u)$

_______________________________________________________

Similarly, defineto writeas a function of

$u = \sin(\underset{v}{ \underbrace{\sin(3x)}}) = \sin(v)$

$u = \sin(v)$

_______________________________________________________

Writeas a function of

$v = \sin(\underset{w}{ \underbrace{3x}} )=\sin(w)$

$v = \sin(w)$

_______________________________________________________

最后,定义最内层函数,, as the function of

w=3x

________________________________________________________

$\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dv}\frac{dv}{dw}\frac{dw}{dx}$

Since $\frac{dw}{dx}=3$ we will just substitute that in and move to the front.

$\frac{dy}{dx}=3\cos(u)\cos(v)\cos(w)$

That was easy enough, now just write everything in terms ofby going back to the definitions ofand.

$u = \sin(\sin(3x))$

$v = \sin(3x)$

w = 3x

$\frac{dy}{dx}=3\cos(\sin(\sin(3x)))\times \cos(\sin(3x))\times \cos(3x)$

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Example Question #371 :Calculus Ab

找到切线。Given the point (1,2)

y= x^2-3x+4

Possible Answers:

y=-x+3

y=x-2

y=x-6

y= x+2

y=x-4

Correct answer:

y=-x+3

Explanation:

To find the tangent line at the given point, we need to first take the derivative of the given function.

Power Rule:

The Power Rule says that we take the exponent of the “x” value and bring it to the front. Then we subtract one from the exponent.

因此, y= x^2-3x+4 becomes y' = 2x - 3

From there we plug in the "1" from the point to get our m value of the equation y=mx + b . When we plug in "1" to y' we get m=-1. Then from there, we will plug our point intonow that we have found m to find our b value. So,

2=(-1)(1) + b

b= 3

因此,the tangent line is equal to

y= -x+3

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Example Question #372 :Calculus Ab

Find the line tangent at the point (0,1)

$y=e^{3x}$

Possible Answers:

y=3x-2

y=3x+3

y=3x+1

y=-3x+2

y=3x+2

Correct answer:

y=3x+1

Explanation:

To find the tangent line at the given point, we need to first take the derivative of the given function. The rule for functions with "e" in it says that the derivative of e^x=e^x However with this function there is also a 3 in the exponent so we will also use chain rule. Chain Rules states that we work from the outside to the inside. Meaning we will take the derivative of the outside of the equation and multiply it by the derivative of the inside of the equation.

To put this into equation it will look like $3e^{3x}$

From there we plug in the "0" from the point to get our m value of the equation . When we plug in "0" to y' we get m=3. Then from there, we will plug our point into y=mx + b now that we have found m to find our m value. So,

1=3(0)+b

b=1 then plug this all back into the equation once more and we are left with

y=3x + 1

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Example Question #373 :Calculus Ab

Find the tangent line given the point (2,4) and the equation

y=2x^2 + 4x

Possible Answers:

y=-8x+12

y=-8x-12

y=8x+12

y=8x-12

y=28x-16

Correct answer:

y=8x-12

Explanation:

To find the tangent line at the given point, we need to first take the derivative of the given function using Power Rule

Power rule says that we take the exponent of the “x” value and bring it to the front. Then we subtract one from the exponent. $(nx^{^{n-1}})$

From there we plug in the "2" from the point to get our m value of the equation. When we plug in "2" to y' we get m=8. Then from there, we will plug our point into now that we have found m to find our b value. So,

y=mx+b

4=8(2) + b ; b=-12

Plug this back into y=mx+b

y= 8x - 12

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Example Question #374 :Calculus Ab

Find the equation of the line tangent to the curve at the point where $x = \frac{\pi}{4}$

$f(x)=\tan(x)+x$

Possible Answers:

$y = 3x - \frac{3\pi}{4}$

$y = \sqrt{2}x+\frac{\pi}{4}$

$y = x+\frac{4-\pi}{2}$

$y = \sqrt{3}x+\pi$

$y = 3x+\frac{2-\pi}{2}$

Correct answer:

$y = 3x+\frac{2-\pi}{2}$

Explanation:

Find the equation of the line tangent to the curveat the given point

$f(x)=\tan(x)+x$ $x = \frac{\pi}{4}$

The slope of the line tangent at the given point will be equal to the derivative of f(x) at that point. Compute the derivative and find the slope for our line:

$f'(x)=\sec^2(x)+1$

$f'\left(\frac{\pi}{4} \right )=\sec^2\left(\frac{\pi}{4}\right)+1$

Evaluate the secant term:

$\sec^2\left(\frac{\pi}{4}\right)=\frac{1}{\cos^2\left(\frac{\pi}{4}\right)}=\left(\frac{2}{\sqrt{2}} \right )^2=2$

Therefore slope of the tangent line is simply:

$f'\left(\frac{\pi}{4} \right )=3$

So now we know the slope of the tangent line and can write the equation then solve for

y = 3x+b

In order to solve forwe need one point on the line. Use the point where the tangent line meets the curve. Use the original function to find the "y" coordinate at this point:

$f\left(\frac{\pi}{4} \right )\: \: =\: \: \tan\left(\frac{\pi}{4}\right) +\frac{\pi}{4}\: \:=\: \: 1+\frac{\pi}{4}\: \: =\: \: \frac{4+\pi}{4}$

We now have our point:

$\left(\frac{\pi}{4}\: \: ,\: \frac{4+\pi}{4}\right )$

Use the point to find

$\frac{4+\pi}{4}=3\left(\frac{\pi}{4} \right )+b$

$b = \frac{2-\pi}{2}$

$y = 3x+\frac{2-\pi}{2}$

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Example Question #375 :Calculus Ab

发现l的斜率ine tangent to the curve of $d\left ( g \right )$ when g=6 .

d(g)=14g^3+6g^2-2g+12

Possible Answers:

d'(g)=84g+12

d'(g)=42g^2+12g-2g+12

d'(6)=1582

d'(6)=1486

Correct answer:

d'(6)=1582

Explanation:

发现l的斜率ine tangent to the curve of d(g) when g=6.

d(g)=14g^3+6g^2-2g+12

All we need here is the power rule. This states that to find the derivative of a polynomial, simply subtract 1 from each exponent and then multiply each term by their original exponent.

Constant terms will drop out when we do this, and linear terms will become constants.

d(g)=14g^3+6g^2-2g+12

d'(g)=(3)14g^2+(2)6g-2=42g^2+12g-2

d'(g)=42g^2+12g-2

From here substitute in g=6.

$\\d'(6)=42(6)^2+12(6)-2 \\d'(6)=1582$

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Example Question #51 :Contextual Applications Of Derivatives

Give the equation of the line tangent to the graph of the equation

$f(x) = 4x+ e^{x-2}$

at the point (2, 9) .

Possible Answers:

y=4x+1

y = 5x-1

y= (4+e)x+(1-2e)

y= (4-e)x+(1+2e)

y=6x-3

Correct answer:

y = 5x-1

Explanation:

The tangent line to the graph ofat point (2, 9) is the line with slope f'(2) that passes through that point. Find the derivative f'(x) :

$f(x) = 4x+ e^{x-2}$

$f(x) = 4x+ \frac{1}{e^{2}} \cdot e^{x}$

$f'(x) =\frac{\mathrm{d} }{\mathrm{d} x} \left (4x+ \frac{1}{e^{2}} \cdot e^{x} \right )$

Apply the sum rule:

$f'(x) =\frac{\mathrm{d} }{\mathrm{d} x} (4x ) + \frac{\mathrm{d} }{\mathrm{d} x} \left ( \frac{1}{e^{2}} \cdot e^{x} \right )$

$f'(x) =\frac{\mathrm{d} }{\mathrm{d} x} (4x ) + \frac{1}{e^{2}} \frac{\mathrm{d} }{\mathrm{d} x} \left ( e^{x} \right )$

$f'(x) =4 + \frac{1}{e^{2}} e^{x}$

$f'(x) =4 + e^{x-2}$

$f'(2)= 4 + e^{2-2} = 4 + e^{0} = 4 + 1 = 5$

The tangent line is therefore the line with slope 5 through (2, 9) . Apply the point-slope formula:

y - 9 = 5(x-2)

y - 9 =5x-10

y = 5x-1

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Example Question #377 :Calculus Ab

Give the equation of the line tangent to the graph of the equation

$f(x) =3 \sin \frac{x}{2} +2 \cos \frac{x}{3}$

at the point (0, 2) .

Possible Answers:

$y =- \frac{2}{3}x+2$

None of the other choices gives the correct response.

$y = -\frac{5}{6}x+2$

$y = \frac{5}{6}x+2$

$y = \frac{2}{3}x+2$

Correct answer:

None of the other choices gives the correct response.

Explanation:

The tangent line to the graph ofat point (0, 2) is the line with slope f'(0) that passes through that point. Find the derivative f'(x) :

$f(x) =3 \sin \frac{x}{2} +2 \cos \frac{x}{3}$

$f'(x) =\frac{\mathrm{d} }{\mathrm{d} x}\left (3 \sin \frac{x}{2} +2 \cos \frac{x}{3} \right )$

Apply the constant multiple and sum rules:

$f'(x) =\frac{\mathrm{d} }{\mathrm{d} x}\left (3 \sin \frac{x}{2} +2 \cos \frac{x}{3} \right )$

$f'(x) =\frac{\mathrm{d} }{\mathrm{d} x}\left (3 \sin \frac{x}{2} \right )+ \frac{\mathrm{d} }{\mathrm{d} x}\left ( 2 \cos \frac{x}{3} \right )$

$f'(x) =3 \frac{\mathrm{d} }{\mathrm{d} x}\left ( \sin \frac{x}{2} \right )+2 \frac{\mathrm{d} }{\mathrm{d} x}\left ( \cos \frac{x}{3} \right )$

Set $u = \frac{x}{2}$ and $v= \frac{x}{3}$ and apply the chain rule.

$f'(x) =3 \frac{\mathrm{d} }{\mathrm{d} x}\left ( \sin u \right )+2 \frac{\mathrm{d} }{\mathrm{d} x}\left ( \cos v \right )$

$f'(x) =3 \frac{\mathrm{d} u}{\mathrm{d} x}\cdot \frac{\mathrm{d} }{\mathrm{d} u} \left ( \sin u \right )+2 \frac{\mathrm{d} v}{\mathrm{d} x}\cdot \frac{\mathrm{d} }{\mathrm{d} v} \left ( \cos v \right )$

$f'(x) =3 \frac{\mathrm{d} u}{\mathrm{d} x}\cdot \cos u +2 \frac{\mathrm{d} v}{\mathrm{d} x}\cdot (- \sin v)$

Substituting back:

$f'(x) =3 \frac{\mathrm{d} }{\mathrm{d} x} \left ( \frac{x}{2} \right )\cdot \cos \frac{x}{2} -2 \frac{\mathrm{d} }{\mathrm{d} x}\left ( \frac{x}{3} \right )\cdot \sin \frac{x}{3}$

$f'(x) =3 \left ( \frac{1}{2} \right )\cdot \cos \frac{x}{2} -2 \left ( \frac{1}{3} \right )\cdot \sin \frac{x}{3}$

$f'(x) = \frac{3}{2} \cos \frac{x}{2} - \frac{2}{3} \sin \frac{x}{3}$

Evaluate f'(0) using substitution:

$f'(0) = \frac{3}{2} \cos \frac{0}{2} - \frac{2}{3} \sin \frac{0}{3}$

$f'(0) = \frac{3}{2} \cos 0 - \frac{2}{3} \sin 0$

$f'(0) = \frac{3}{2}(1) - \frac{2}{3}( 0)$

$f'(0) = \frac{3}{2}$

The tangent line is therefore the line with slope $\frac{3}{2}$ through (0, 2) . (0, 2) is a-intercept, so apply the slope-intercept formula to get the equation

$y = \frac{3}{2}x+2$ .

This is not among the choices given.

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Example Question #1 :Determine Local Linearity And Linearization

Find the equation of the line parallel to the function y=4x^3+2 at x=1 , and passes through the point (2,4)

Possible Answers:

y=5x-10

y=12x-20

y=15x-2

y=-6x+3

Correct answer:

y=12x-20

Explanation:

We first start by finding the slope of the line in question, which we do by taking the derivative of y=4x^3+2 and evaluate at x=1 .

y'=12x^2 , y'(1)=12=m

We then use point slope form to get the equation of the line at the point (2,4)

(y-4)=12(x-2)

y=12x-20

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Example Question #379 :Calculus Ab

Find the equation of the line tangent to $f(x)=x^{3}-3x^{2}+2x+4$ at the point (1,4) .

Possible Answers:

y=-5x+1

y=-x+5

y=-2x+5

$y=3x^{2}+2x+4$

y=x+1

Correct answer:

y=-x+5

Explanation:

The first step is to find the derivative of the function given, which is $f'(x)=3x^{2}-6x+2$ . Next, find the slope at (1,4) by plugging in x=1 and solving for f'(x) , which is the slope. You should get f'(1)= -1 . This means the slope of the new line is also -1 because at the point where a slope and a line are tangent they have the same slope. Use the equation y=mx+b to express your line. Y and x are variables and m is the slope, so the only thing you need to find is b. Plug in the point and slope into y=mx+b to get b=5 . Now you can express the general equation of the line as y=-x+5 .

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微积分AB:确定当地的线性和线性ization

Study concepts, example questions & explanations for Calculus AB

All Calculus AB Resources

Example Questions

Example Question #1 :Determine Local Linearity And Linearization

Example Question #371 :Calculus Ab

Example Question #372 :Calculus Ab

Example Question #373 :Calculus Ab

Example Question #374 :Calculus Ab

Example Question #375 :Calculus Ab

Example Question #51 :Contextual Applications Of Derivatives

Example Question #377 :Calculus Ab

Example Question #1 :Determine Local Linearity And Linearization

Example Question #379 :Calculus Ab

All Calculus AB Resources

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