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Calculus 1 : Other Differential Functions

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Example Question #41 : How To Find Differential Functions

Given:

$Lim _{n \rightarrow \infty} \left (1+\frac{1}{n} \right )^n=e$

Evaluate the limit:

$Lim _{n \rightarrow \infty} \left (1+\frac{1}{n} \right )^{2n}$

Possible Answers:

DoesNotExist

e^2

$\frac{e}{2}$

Correct answer:

e^2

Explanation:

Recalling properties of exponents:

$Lim _{n \rightarrow \infty} \left (1+\frac{1}{n} \right )^{2n}=Lim _{n \rightarrow \infty}\left ( \left (1+\frac{1}{n} \right )^{n} \right )^2=Lim _{n \rightarrow \infty} \left (1+\frac{1}{n} \right )^{n}\left (1+\frac{1}{n} \right )^{n}$

Limit of product is the product of the limits:

$Lim _{n \rightarrow \infty} \left (1+\frac{1}{n} \right )^{n}\left (1+\frac{1}{n} \right )^{n}=Lim _{n \rightarrow \infty} \left (1+\frac{1}{n} \right )^{n} Lim _{n \rightarrow \infty} \left (1+\frac{1}{n} \right )^{n}$

$= \left (Lim _{n \rightarrow \infty} \left (1+\frac{1}{n} \right )^{n} \right )^2$

And from the Pre-Question Text:

$Lim _{n \rightarrow \infty} \left (1+\frac{1}{n} \right )^n=e$

So:

$\left (Lim _{n \rightarrow \infty} \left (1+\frac{1}{n} \right )^{n} \right )^2 = e^2$

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Example Question #42 : How To Find Differential Functions

Given:

$Lim _{n \rightarrow \infty} \left (1+\frac{1}{n} \right )^n=e$

Evaluate the limit:

$Lim _{n \rightarrow 0} \left (1+n \right )^{\frac{1}{n}}$

Possible Answers:

$\frac{1}{e}$

DoesNotExist

Correct answer:

Explanation:

To solve this problem we use the variable substitution:

$m = \frac{1}{n}$

Obtaining:

$Lim _{n \rightarrow 0} \left (1+n \right )^{\frac{1}{n}}=Lim _{m \rightarrow \infty} \left (1+\frac{1}{m} \right )^{m}$

And from the Pre-Question Text:

$Lim _{n \rightarrow \infty} \left (1+\frac{1}{n} \right )^n=e$

So:

$Lim _{m \rightarrow \infty} \left (1+\frac{1}{m} \right )^{m}=e$

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Example Question #43 : How To Find Differential Functions

Given:

$Lim _{n \rightarrow \infty} \left (1+\frac{1}{n} \right )^n=e$

Evaluate the limit:

$Lim _{n \rightarrow \infty} \left (1+\frac{1}{2n} \right )^{n}$

Possible Answers:

$\frac{e}{2}$

DoesNotExist

$\sqrt[]{e}$

Correct answer:

$\sqrt[]{e}$

Explanation:

To solve this problem we use the variable substitution:

m = 2n

Obtaining:

$Lim _{n \rightarrow \infty} \left (1+\frac{1}{2n} \right )^{n} = Lim _{m \rightarrow \infty} \left (1+\frac{1}{m} \right )^{\frac{m}{2}} = L$

We then observe that:

$L^2 =Lim _{m \rightarrow \infty} \left (1+\frac{1}{m} \right )^{\frac{m}{2}}*Lim _{m \rightarrow \infty} \left (1+\frac{1}{m} \right )^{\frac{m}{2}}$

Product of limits is the limit of the products:

$= Lim _{m \rightarrow \infty} \left (1+\frac{1}{m} \right )^\frac{m}{2}*\left (1+\frac{1}{m} \right )^\frac{m}{2}=Lim _{m \rightarrow \infty}\left (1+\frac{1}{m} \right )^m$

And from the Pre-Question Text:

$Lim _{n \rightarrow \infty} \left (1+\frac{1}{n} \right )^n=e$

So:

$L^2 = Lim _{m \rightarrow \infty}\left (1+\frac{1}{m} \right )^m=e$

$L^2 = e \rightarrow L=e^{\frac{1}{2}}=\sqrt[]{e}$

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Example Question #44 : How To Find Differential Functions

Given:

$Lim _{x \rightarrow 0} \frac{sinx}{x} = 1$

$Lim _{x \rightarrow 0} \frac{1 - cosx}{x} = 0$

Evaluate the limit:

$Lim _{x \rightarrow 0} \frac{x-xcosx}{sin^{2}x}$

Possible Answers:

DoesNotExist

$\pi^2$

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

Correct answer:

Explanation:

Multiplying by $\frac{x}{x}$ we obtain:

$Lim _{x \rightarrow 0} \frac{x*(x-xcosx)}{x*sin^{2}x} = Lim _{x \rightarrow 0} \frac{x}{sinx}*\frac{x}{sinx}* \frac{1-cosx}{x}$

Limit of product is the product of limits:

$Lim _{x \rightarrow 0} \frac{x}{sinx}*\frac{x}{sinx}* \frac{1-cosx}{x} = Lim _{x \rightarrow 0} \frac{x}{sinx}*Lim _{x \rightarrow 0}\frac{x}{sinx}*Lim _{x \rightarrow 0} \frac{1-cosx}{x}$

And from the Pre-Question Text:

$Lim _{x \rightarrow 0} \frac{sinx}{x} = 1$ and $Lim _{x \rightarrow 0} \frac{1 - cosx}{x} = 0$

So:

$Lim _{x \rightarrow 0} \frac{x}{sinx}*Lim _{x \rightarrow 0}\frac{x}{sinx}*Lim _{x \rightarrow 0} \frac{1-cosx}{x}=1*1*0=0$

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Example Question #45 : How To Find Differential Functions

Given:

$Lim _{x \rightarrow 0} \frac{sinx}{x} = 1$

$Lim _{x \rightarrow 0} \frac{1 - cosx}{x} = 0$

Evaluate the limit:

$Lim _{x \rightarrow 0} \frac{csc7x}{csc2x}$

Possible Answers:

Does Not Exist

$\frac{7}{2}$

$\frac{1}{14}$

$\frac{2}{7}$

Correct answer:

$\frac{2}{7}$

Explanation:

First observe that $\frac{csc7x}{csc2x} = \frac{sin2x}{sin7x}$

Multiplying by $\frac{2x}{2x}$ and $\frac{7x}{7x}$ we obtain:

$Lim _{x \rightarrow 0} \frac{2x*7x*sin2x}{2x*7x*sin7x} = Lim _{x \rightarrow 0} \frac{sin2x}{2x}*\frac{7x}{sin7x}* \frac{2x}{7x}$

Limit of product is the product of limits:

$Lim _{x \rightarrow 0} \frac{sin2x}{2x}*\frac{7x}{sin7x}* \frac{2x}{7x}=Lim _{x \rightarrow 0} \frac{sin2x}{2x}*Lim _{x \rightarrow 0}\frac{7x}{sin7x}*Lim _{x \rightarrow 0} \frac{2x}{7x}$

And from the Pre-Question Text:

$Lim _{x \rightarrow 0} \frac{sinx}{x} = 1$

So:

$Lim _{x \rightarrow 0} \frac{sin2x}{2x}*Lim _{x \rightarrow 0}\frac{7x}{sin7x}*Lim _{x \rightarrow 0} \frac{2x}{7x}=1*1*\frac{2}{7}=\frac{2}{7}$

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Example Question #46 : How To Find Differential Functions

Given:

$Lim _{x \rightarrow 0} \frac{sinx}{x} = 1$

$Lim _{x \rightarrow 0} \frac{1 - cosx}{x} = 0$

Evaluate the limit:

$Lim _{x \rightarrow 0} \frac{cos5x-1}{2x}$

Possible Answers:

$\frac{2\pi }{5}$

Does Not Exist

$\frac{2}{5}$

$\frac{5}{2}$

Correct answer:

Explanation:

Multiplying by $\frac{5x}{5x}$ we obtain:

$Lim _{x \rightarrow 0} \frac{5x*(cos5x-1)}{5x*2x} = Lim _{x \rightarrow 0} \frac{5x}{2x}*\frac{cos5x-1}{5x}$

Limit of product is the product of limits:

$Lim _{x \rightarrow 0} \frac{5x}{2x}*\frac{cos5x-1}{5x}=Lim _{x \rightarrow 0} \frac{5x}{2x}*Lim _{x \rightarrow 0}\frac{cos5x-1}{5x}$

And from the Pre-Question Text:

$Lim _{x \rightarrow 0} \frac{1 - cosx}{x} = 0$

So:

$Lim _{x \rightarrow 0} \frac{5x}{2x}*Lim _{x \rightarrow 0}\frac{cos5x-1}{5x}=\frac{5}{2}*0=0$

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Example Question #231 : Differential Functions

Given:

$Lim _{x \rightarrow 0} \frac{sinx}{x} = 1$

$Lim _{x \rightarrow 0} \frac{1 - cosx}{x} = 0$

Evaluate the limit:

$Lim _{x \rightarrow 0} \frac{sin2x}{3x}$

Possible Answers:

Does Not Exist

$\frac{2}{3}$

$\frac{2\pi }{3}$

$\frac{3}{2}$

Correct answer:

$\frac{2}{3}$

Explanation:

Multiplying by $\frac{2x}{2x}$ we obtain:

$Lim _{x \rightarrow 0} \frac{2x*sin2x}{2x*3x} = Lim _{x \rightarrow 0} \frac{2x}{3x}*\frac{sin2x}{2x}$

Limit of product is the product of limits:

$Lim _{x \rightarrow 0} \frac{2x}{3x}*\frac{sin2x}{2x}=Lim _{x \rightarrow 0} \frac{2x}{3x}*Lim _{x \rightarrow 0}\frac{sin2x}{2x}$

And from the Pre-Question Text:

$Lim _{x \rightarrow 0} \frac{sinx}{x} = 1$

So:

$Lim _{x \rightarrow 0} \frac{2x}{3x}*Lim _{x \rightarrow 0}\frac{sin2x}{2x}=\frac{2}{3}*1=\frac{2}{3}$

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Example Question #48 : How To Find Differential Functions

Differentiate the function using known derivatives and applying the product, quotient, and chain rules.

f(x)=x*cos(2x)

Possible Answers:

f'(x)=cos(2x)-2x*sin(2x)

f(x)=-2sin(2x)

f'(x)=cos(2x)+2x*sin(2x)

f'(x)=cos(2x)+2*sin(2x)

f'(x)=sin(2x)-2x*cos(2x)

Correct answer:

f'(x)=cos(2x)-2x*sin(2x)

Explanation:

f(x)=x*cos(2x)

Using the product rule, $if \ f(x)=g(x) *h(x)$ ,

$then\ f'(x)=g'(x)*h(x) + g(x)*h'(x)$ ,

we observe the following:

$g(x)= x \ so\ g'(x)= 1$

$h(x) = cos(2x) \ so \ h'(x)=-2sin(2x) \ [This\ derivative \ uses\ the \ chain \ rule.]$

f'(x)= 1*cos(2x) + x*(-2sin(2x)) = cos(2x)-2xsin(2x)

f'(x)=cos(2x)-2xsin(2x) , which is our final answer.

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Example Question #49 : How To Find Differential Functions

Differentiate the function using known derivatives and applying the product, quotient, and chain rules.

$f(x)=sin^{2}(x) + cos^{2}(x)$

Possible Answers:

f'(x)=1

f'(x)=2cos(x) - 2sin(x)

f'(x)=4sin(x)*cos(x)

f'(x)=0

f'(x)=2sin(x) + 2cos(x)

Correct answer:

f'(x)=0

Explanation:

$f(x)=sin^{2}(x) + cos^{2}(x)$

We will need to use either the chain rule or product rule on both summands. The chain rule involves less simplifiction at the end. It states: $if \ f(x)=g(h(x))$ , $then\ f'(x)=g'(h(x))*h'(x)$ .

In this case, the outside function is, in fact, x^2 in both cases.

$g(x)= x^2 \ so\ g'(x)= 2x$

For the first summand:

$h(x) = sin(x) \ so \ h'(x)= cos(x)$

And for the the second:

$h(x) = cos(x) \ so \ h'(x)= -sin(x)$

f'(x)= 2sin(x)*cos(x) - 2cos(x)*sin(x) = 0

f'(x)=0 , which is our final answer.

[A function with a derivative of zero is a constant. You might recall the trig identity:

$sin^{2}(x) + cos^{2}(x) = 1$ , which is another way to solve this problem.]

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Example Question #50 : How To Find Differential Functions

Differentiate the function using known derivatives and applying the product, quotient, and chain rules.

f(x)=(x+1)(x+2)(x+3)

Possible Answers:

f'(x)=(x+2)(x+3) + 2(x+1)(x+3) + 3(x+1)(x+2)

f'(x)=(x+2)(x+3) + (x+1)(x+3) + (x+1)(x+2)

f'(x)=3x+6

f'(x)=1

f'(x)=x^3

Correct answer:

f'(x)=(x+2)(x+3) + (x+1)(x+3) + (x+1)(x+2)

Explanation:

f(x)=(x+1)(x+2)(x+3)

You could expand this problem out, but let's evaluate the derivative using a modified product rule:

$if \ f(x)=g(x)*h(x)*j(x)$

$then\ f'(x)=g'(x)*h(x)*j(x)+g(x)*h'(x)*j(x)+g(x)*h(x)*j'(x)$ .

Setting each of the multiplicands as , , and , respectively, we get:

f'(x)=1*(x+2)(x+3) + (x+1)*1*(x+3) + (x+1)(x+2)*1

f'(x)=(x+2)(x+3) + (x+1)(x+3) + (x+1)(x+2) , which is our final answer.

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Calculus 1 : Other Differential Functions

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #41 : How To Find Differential Functions

Example Question #42 : How To Find Differential Functions

Example Question #43 : How To Find Differential Functions

Example Question #44 : How To Find Differential Functions

Example Question #45 : How To Find Differential Functions

Example Question #46 : How To Find Differential Functions

Example Question #231 : Differential Functions

Example Question #48 : How To Find Differential Functions

Example Question #49 : How To Find Differential Functions

Example Question #50 : How To Find Differential Functions

All Calculus 1 Resources

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