Calculus 1 : Functions

Study concepts, example questions & explanations for Calculus 1

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Example Questions

Example Question #464 : How To Find Differential Functions

Find the slope of the function \(\displaystyle f(x,y)=tan(x)ln(y)\) at \(\displaystyle (\pi,e)\).

Possible Answers:

\(\displaystyle (-1,\frac{1}{e})\)

\(\displaystyle (1,0)\)

\(\displaystyle (1,\frac{1}{e})\)

\(\displaystyle (-1,0)\)

\(\displaystyle (1,e)\)

Correct answer:

\(\displaystyle (1,0)\)

Explanation:

To consider finding the slope, let's discuss the topic of the gradient.

For a function \(\displaystyle f(x_1,x_2,...,x_n)\), the gradient is the sum of the derivatives with respect to each variable, multiplied by a directional vector:

\(\displaystyle \frac{\delta f}{\delta x_1}\widehat{e_1}+\frac{\delta f}{\delta x_2}\widehat{e_2}+...+\frac{\delta f}{\delta x_n}\widehat{e_n}\)

It is essentially the slope of a multi-dimensional function at any given point

Knowledge of the following derivative rules will be necessary:

Derivative of a natural log: 

\(\displaystyle d[ln(u)]=\frac{du}{u}\)

Trigonometric derivative: 

\(\displaystyle d[tan(u)]=\frac{du}{cos^2(u)}\)

Note that u may represent large functions, and not just individual variables!

The approach to take with this problem is to simply take the derivatives one at a time. When deriving for one particular variable, treat the other variables as constant.

\(\displaystyle f(x,y)=tan(x)ln(y)\) at \(\displaystyle (\pi,e)\)

x:

\(\displaystyle \frac{\delta f}{\delta x}=\frac{ln(y)}{cos^2(x)};\frac{ln(e)}{cos^2(\pi)}=1\)

y:

\(\displaystyle \frac{\delta f}{\delta y}=\frac{tan(x)}{y};\frac{tan(\pi)}{e}=0\)

The slope is \(\displaystyle (1,0)\)

Example Question #465 : How To Find Differential Functions

Find the derivative.

\(\displaystyle 2x^6-3x^3+2x\)

Possible Answers:

\(\displaystyle 12x^5-9x^2-2\)

\(\displaystyle 12x-9x^2+2\)

\(\displaystyle 9x^5-9x^2+2\)

\(\displaystyle 12x^5-9x^2+2\)

Correct answer:

\(\displaystyle 12x^5-9x^2+2\)

Explanation:

Use the power rule to find the derivative.

Remember the power rule is:

\(\displaystyle \\ \\ f(x)=ax^n \\ f'(x)=nax^{n-1}\)

 

\(\displaystyle \frac{d}{dx}2x^6=12x^5\)

\(\displaystyle \frac{d}{dx}-3x^3=-9x^2\)

\(\displaystyle \frac{d}{dx}2x=2\)

Thus, the derivative is \(\displaystyle 12x^5-9x^2+2\)

Example Question #466 : How To Find Differential Functions

Find the derivative at \(\displaystyle x=9\).

\(\displaystyle 4x-5\)

Possible Answers:

\(\displaystyle 5\)

\(\displaystyle 7\)

\(\displaystyle 4\)

\(\displaystyle 6\)

Correct answer:

\(\displaystyle 4\)

Explanation:

First, find the derivative using the power rule

Recall the power rule:

\(\displaystyle \\ \\ f(x)=ax^n \\ f'(x)=nax^{n-1}\)

\(\displaystyle \frac{d}{dx}4x=4\)

\(\displaystyle \frac{d}{dx}-5=0\)

Because the derivative is a constant \(\displaystyle 4\), then the derivative at all points is \(\displaystyle 4\).

Example Question #467 : How To Find Differential Functions

Find the derivative at \(\displaystyle x=0\).

\(\displaystyle x^2+9x-3\)

Possible Answers:

\(\displaystyle 9\)

\(\displaystyle 7\)

\(\displaystyle 12\)

\(\displaystyle 5\)

Correct answer:

\(\displaystyle 9\)

Explanation:

First, find the derivative using the power rule. 

Recall the power rule:

\(\displaystyle \\ \\ f(x)=ax^n \\ f'(x)=nax^{n-1}\)

\(\displaystyle \frac{d}{dx}x^2=2x\)

\(\displaystyle \frac{d}{dx}9x=9\)

Recall that the derivative of a constant is zero.

The derivative is \(\displaystyle 2x+9.\)

Now, substitute \(\displaystyle 0\) for \(\displaystyle x\).

\(\displaystyle 2(0)+9\)

\(\displaystyle =9\)

Example Question #651 : Functions

Find the derivative. 

\(\displaystyle x^2\sin(x)\)

Possible Answers:

\(\displaystyle \cos (x)+\sin (x)\)

\(\displaystyle x^2\cos (x)-2x\sin (x)\)

\(\displaystyle x\cos (x)+2x\sin (x)\)

\(\displaystyle x^2\cos (x)+2x\sin (x)\)

Correct answer:

\(\displaystyle x^2\cos (x)+2x\sin (x)\)

Explanation:

Uses the product rule to find this derivative.

Recall the product rule:

\(\displaystyle \\ \\ f(x)=(f(x)g(x))' \\ f'(x)=f'(x)g(x)+f(x)g'(x)\)

\(\displaystyle f'(x)=2x\sin (x)+x^2\cos (x)\)

Example Question #652 : Functions

Find the derivative.

\(\displaystyle 5x\cos (x)\)

Possible Answers:

\(\displaystyle 5x\sin (x)+5\cos (x)\)

\(\displaystyle 5\cos (x)-5x\sin (x)\)

\(\displaystyle -5x\sin (x)-5\cos (x)\)

\(\displaystyle -5x\cos (x)+5\sin (x)\)

Correct answer:

\(\displaystyle 5\cos (x)-5x\sin (x)\)

Explanation:

Use the product rule to find this derivative.

Recall the product rule:

\(\displaystyle \\ \\ f(x)=(f(x)g(x))' \\ f'(x)=f'(x)g(x)+f(x)g'(x)\)

\(\displaystyle f'(x)=5\cos (x)-5x\sin (x)\)

Example Question #470 : How To Find Differential Functions

Find the slope of the tangent line at \(\displaystyle x=8\).

\(\displaystyle 5x^2-7\)

Possible Answers:

\(\displaystyle 90\)

\(\displaystyle 85\)

\(\displaystyle 75\)

\(\displaystyle 80\)

Correct answer:

\(\displaystyle 80\)

Explanation:

First, find the derivative using the power rule. 

Recall the power rule:

\(\displaystyle \\ \\ f(x)=ax^n \\ f'(x)=nax^{n-1}\)

\(\displaystyle \frac{d}{dx}5x^2=10x\)

\(\displaystyle \frac{d}{dx}-7=0\)

The derivative is \(\displaystyle 10x\).

Now, substitute \(\displaystyle 8\) for \(\displaystyle x\).  The slope of the tangent line at \(\displaystyle x=8\) is \(\displaystyle 80\).

Example Question #471 : Other Differential Functions

Find the derivative.

\(\displaystyle -\frac{2}{x^4}\)

Possible Answers:

\(\displaystyle -\frac{4}{x^5}\)

\(\displaystyle -\frac{8}{x^5}\)

\(\displaystyle \frac{4}{x^5}\)

\(\displaystyle \frac{8}{x^5}\)

Correct answer:

\(\displaystyle \frac{8}{x^5}\)

Explanation:

Use the quotient rule to find this derivative.

Recall the quotient rule:

\(\displaystyle \\ \\ h(x)=\frac{f(x)}{g(x)} \\ \\ h'(x)=\frac{g(x)f'(x)-g'(x)f(x)}{(g(x))^2}\)

\(\displaystyle h'(x)=\frac{x^4(0)-2(4x^3)}{-x^8}\)

\(\displaystyle =\frac{-8x^3}{-x^8}=\frac{8x^3}{x^8}=\frac{8}{x^5}\)

Example Question #472 : Other Differential Functions

Find the derivative at \(\displaystyle x=7\).

\(\displaystyle 3x^2-2x^2+5x-4\)

Possible Answers:

\(\displaystyle 19\)

\(\displaystyle 17\)

\(\displaystyle 21\)

\(\displaystyle 15\)

Correct answer:

\(\displaystyle 19\)

Explanation:

Begin by simplifying the expression.

\(\displaystyle x^2+5x-4\)

Now, find the derivative using the power rule.

Recall the power rule:

\(\displaystyle \\ \\ f(x)=ax^n \\ f'(x)=nax^{n-1}\)

\(\displaystyle 2x+5\)

Finally, substitute \(\displaystyle 7\) for \(\displaystyle x\).

\(\displaystyle 2(7)+5=19\)

Example Question #473 : Other Differential Functions

Find the slope of the tangent line at \(\displaystyle x=6\).

\(\displaystyle x^3\)

Possible Answers:

\(\displaystyle 36\)

\(\displaystyle 64\)

\(\displaystyle 124\)

\(\displaystyle 108\)

Correct answer:

\(\displaystyle 108\)

Explanation:

First, find the derivative using the power rule. 

Recall the power rule:

\(\displaystyle \\ \\ f(x)=ax^n \\ f'(x)=nax^{n-1}\)

\(\displaystyle \frac{d}{dx}x^3=3x^2\)

Now, substitute \(\displaystyle 6\) for \(\displaystyle x\).

\(\displaystyle 3(6^2)=3(36)=108\)

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