Calculus 1 : Differential Functions

Study concepts, example questions & explanations for Calculus 1

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Example Questions

Example Question #22 : Other Differential Functions

Integrate

\(\displaystyle \small \int\sin^2x\hspace2dx\)

Possible Answers:

\(\displaystyle \small \small \small \frac{x}{2}+\frac{\cos2x}{4}+C\)

\(\displaystyle \small \small \frac{x}{2}+\frac{\sin2x}{4}+C\)

\(\displaystyle \small \small \small \frac{x}{2}+\frac{\sin2x}{4}\)

\(\displaystyle \small \small \small \frac{x}{2}-\frac{\cos2x}{4}+C\)

\(\displaystyle \small \frac{x}{2}-\frac{\sin2x}{4}+C\)

Correct answer:

\(\displaystyle \small \frac{x}{2}-\frac{\sin2x}{4}+C\)

Explanation:

We can use trigonometric identities to transform integrals that we typically don't know how to integrate. 


\(\displaystyle \small \small \sin^2x = \frac{1-\cos2x}{2}\)
Thus,
\(\displaystyle \small \small \small \int\sin^2x\hspace2dx = \int \frac{1-\cos2x}{2}\hspace2dx\)
\(\displaystyle \small = \frac{1}{2} \int (1-\cos2x) \hspace2 dx = \frac{1}{2}x-\frac{1}{2}\frac{sin2x}{2}+C = \frac{x}{2} - \frac{sin2x}{4}+C\)

Example Question #23 : Other Differential Functions

Integrate

 \(\displaystyle \small \int \cos^2x \hspace2 dx\)

Possible Answers:

\(\displaystyle \small \small \frac{x}{2}+\frac{sin2x}{4}\)

\(\displaystyle \small \small \frac{x}{2}-\frac{sin2x}{4}+C\)

\(\displaystyle \small \small \frac{x}{2}-\frac{cos2x}{4}+C\)

\(\displaystyle \small \small \frac{x}{2}+\frac{cos2x}{4}+C\)

\(\displaystyle \small \frac{x}{2}+\frac{sin2x}{4}+C\)

Correct answer:

\(\displaystyle \small \frac{x}{2}+\frac{sin2x}{4}+C\)

Explanation:

We can use trigonometric identities to integrate functions we typically don't know how to integrate. 


\(\displaystyle \small \cos^2x = \frac{1+\cos2x}{2}\)

Thus,

\(\displaystyle \small \small \int \cos^2x \hspace2 dx = \int \frac{1+\cos2x}{2}\hspace2dx = \frac{x}{2}+\frac{\sin2x}{4}+C\)

Example Question #24 : Other Differential Functions

Evaluate

 \(\displaystyle \small \int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \cot \theta \csc^2 \theta \hspace2 d\theta\)

Possible Answers:

\(\displaystyle \small \frac{1}{2}\)

\(\displaystyle 1\)

\(\displaystyle \small -\frac{1}{2}\)

\(\displaystyle \small -\frac{\pi}{2}\)

\(\displaystyle \small \frac{\pi}{4}\)

Correct answer:

\(\displaystyle \small \frac{1}{2}\)

Explanation:

You can transform the limits of integration via u-substitution. 

Let \(\displaystyle \small \small u = \cot \theta, du = -\csc^2\theta\hspace2d\theta\)
\(\displaystyle \small -du = \csc^2\theta\hspace2d\theta\)


When \(\displaystyle \small \theta = \frac{\pi }{4}, u = \cot\left ( \frac{\pi }{4}\right )=1\)
When \(\displaystyle \small \theta = \frac{\pi }{2}, u = \cot\left ( \frac{\pi }{2}\right ) = 0\)

Thus,

\(\displaystyle \small \small \int_{\pi/4}^{\pi/2} \cot \theta \csc^2 \theta \hspace2 d\theta = \int_{1}^{0}u\cdot(-du)\)

\(\displaystyle \small \small = -\int_{1}^{0}u\hspace2 du\)
\(\displaystyle \small = -\left [ \frac{u^2}{2}\right ]^0_1\)
\(\displaystyle \small = \left [ \frac{(0)^2}{2} - \frac{(1)^2}{2}\right ] = \frac{1}{2}\)

Example Question #25 : Other Differential Functions

Differentiate the function \(\displaystyle y = x^{2}e^x\)

Possible Answers:

\(\displaystyle y^{'}=2e^x\)

\(\displaystyle y^{'}=xe^x\)

\(\displaystyle y^{'}=2 x^2e^x\)

\(\displaystyle y^{'}=2xe^x\)

\(\displaystyle y^{'}=2 xe^x+x^2e^x\)

Correct answer:

\(\displaystyle y^{'}=2 xe^x+x^2e^x\)

Explanation:

Using the product rule for finding derivatives gives the answer.  \(\displaystyle f^{'}(x)g(x)+g^{'}(x)f(x)\)

Example Question #26 : Other Differential Functions

Solve for \(\displaystyle f{}'(x)\) when \(\displaystyle f(x) = \frac{2x^2-4}{x+3}\)

Possible Answers:

\(\displaystyle f{}'(x)= \frac{4x}{4}\)

\(\displaystyle f{}'(x)=\frac{4x^2+8x-12}{(x+3)^2}\)

\(\displaystyle f{}'(x)=\frac{2x^2+12x+4}{x+3}\)

\(\displaystyle f{}'(x)=\frac{2x^2+12x+4}{(x+3)^2}\)

Correct answer:

\(\displaystyle f{}'(x)=\frac{2x^2+12x+4}{(x+3)^2}\)

Explanation:

\(\displaystyle f(x) = \frac{2x^2-4}{x+3}\)

using the quotient rule: \(\displaystyle (\frac{f}{g}){}'= \frac{f{}'g-fg{}'}{g^2}\)

 

\(\displaystyle f{}'(x)=\frac{(4x)(x+3)-(2x^2-4)(1)}{(x+3)^2}\)

Foil

\(\displaystyle f{}'(x)=\frac{4x^2+12x-2x^2+4}{(x+3)^2}\)

combine like-terms to simplify

\(\displaystyle f{}'(x)=\frac{2x^2+12x+4}{(x+3)^2}\)

Example Question #27 : Other Differential Functions

Solve for \(\displaystyle f{}'(x)\) when \(\displaystyle f(x)= 3tan(x)cot(4x)\)

Possible Answers:

\(\displaystyle f{}'(x)= sec^2(x)cot(4x)-4tan(x)csc^2(4x)\)

\(\displaystyle f{}'(x)= 3sec^2(x)cot(4x)-12tan(x)csc^2(4x)\)

\(\displaystyle f{}'(x)= 3sec^2(x)cot(4x)+12tan(x)csc^2(4x)\)

\(\displaystyle f{}'(x)= 3sec^2(x)cot(4x)-3tan(x)csc^2(4x)\)

Correct answer:

\(\displaystyle f{}'(x)= 3sec^2(x)cot(4x)-12tan(x)csc^2(4x)\)

Explanation:

\(\displaystyle f(x)= 3tan(x)cot(4x)\)

Using the Product Rule: \(\displaystyle (fg){}'=f{}'g+fg{}'\) and chain rule for trignometry functions: \(\displaystyle \frac{d}{dx}(cot[f(x)])=f{}'(x)cot[f(x)]\)

\(\displaystyle f{}'(x)=(3sec^2(x))(cot(4x))+(3tan(x))(4)(-csc^2(4x))\)

Simplify

\(\displaystyle f{}'(x)= 3sec^2(x)cot(4x)-12tan(x)csc^2(4x)\)

Example Question #28 : Other Differential Functions

Find the derivative of \(\displaystyle y = sin((\sqrt{x})^3)\)

Possible Answers:

\(\displaystyle y^{'}=\frac{2\sqrt{x}\cdot sin(x^\frac{3}{2})}{3}\)

\(\displaystyle y^{'}=\frac{\sqrt{x}\cdot sin(x^\frac{3}{2})}{2}\)

\(\displaystyle y^{'}=\frac{3cos(x^\frac{3}{2})}{\sqrt{x}}\)

\(\displaystyle y^{'}=\frac{3\sqrt{x}\cdot cos(x^\frac{3}{2})}{2}\)

\(\displaystyle y^{'}=\frac{3\sqrt{x}\cdot sin(x)}{2}\)

Correct answer:

\(\displaystyle y^{'}=\frac{3\sqrt{x}\cdot cos(x^\frac{3}{2})}{2}\)

Explanation:

The quantity square root of \(\displaystyle x\) raised to the third is the same as \(\displaystyle x^{\frac{3}{2}}\).  Using the chain rule and power rule, the answer can be found.

Example Question #31 : Other Differential Functions

Solve for \(\displaystyle f{}'(x)\) when 

\(\displaystyle f(x)= \left(9x^4+4x^3+\frac{x}{2}\right)^9\)

Possible Answers:

\(\displaystyle f{}'(x)=9(9x^4+4x^3+\frac{x}{2})^8\)

\(\displaystyle f{}'(x)=36x^3+12x^2+\frac{1}{2}\)

\(\displaystyle f{}'(x)=(9x^4+4x^3+\frac{x}{2})^9(36x^3+12x^2+\frac{1}{2})\)

\(\displaystyle f{}'(x)=(9x^4+4x^3+\frac{x}{2})^8(324x^3+108x^2+\frac{9}{2})\)

Correct answer:

\(\displaystyle f{}'(x)=(9x^4+4x^3+\frac{x}{2})^8(324x^3+108x^2+\frac{9}{2})\)

Explanation:

\(\displaystyle f(x)= \left(9x^4+4x^3+\frac{x}{2}\right)^9\)

using the chain rule:

 \(\displaystyle \frac{d}{dx}(f(g(x)))=f{}'(g(x))g{}'(x)\)

\(\displaystyle f{}'(x)=9(9x^4+4x^3+x/2)^8(36x^3+12x^2+1/2)\)

multiply the constant 9 into the second function to simplify answer

Example Question #32 : Other Differential Functions

Solve for \(\displaystyle f{}'(x)\) when \(\displaystyle f(x)=\log_{9}x^2\)

Possible Answers:

\(\displaystyle f{}'(x)= \frac{2}{x}\)

\(\displaystyle f{}'(x)=\frac{2}{x\ln 9}\)

\(\displaystyle f{}'(x)=\frac{x}{\ln 9}\)

\(\displaystyle f{}'(x)=\frac{2}{\ln 9}\)

Correct answer:

\(\displaystyle f{}'(x)=\frac{2}{x\ln 9}\)

Explanation:

\(\displaystyle f(x)=\log_{9}x^2\)

using the logarithm identities change the equation to base 10: 

\(\displaystyle f(x)=\frac{\ln x^2}{\ln 9}\)

using lograthim identities simplify the numerator:

\(\displaystyle f(x)=\frac{2\ln x}{\ln 9}\)

differentiate

\(\displaystyle f{}'(x)=\frac{2*\frac{1}{x}}{\ln 9}\)

Simplify

\(\displaystyle f{}'(x)=\frac{2}{x\ln 9}\)

Example Question #33 : Other Differential Functions

Solve for \(\displaystyle f'(x)\) when

 \(\displaystyle f(x)= (x^2 +7x)(3x^3+\frac{x}{4})\)

Possible Answers:

\(\displaystyle f{}'(x)= 6x^4 +21x^3 +\frac{x^2}{2}+\frac{7x}{4}\)

\(\displaystyle f'(x) = 81x^2 + \frac{x}{2}+\frac{7}{4}\)

\(\displaystyle f{}'(x)= 15x^4 + 21x^3 +255x^2 + 7x +7\)

\(\displaystyle f^{'}(x)=15x^{4}+84x^{3}+\frac{3x^{2}}{4}+\frac{7x}{2}\)

Correct answer:

\(\displaystyle f^{'}(x)=15x^{4}+84x^{3}+\frac{3x^{2}}{4}+\frac{7x}{2}\)

Explanation:

\(\displaystyle f(x)= (x^2 +7x)(3x^3+\frac{x}{4})\)

Using the product rule: \(\displaystyle (fg){}'=f{}'g+fg{}'\)

 

\(\displaystyle f{}'(x)= (2x+7)(3x^3 + x/4) + (x^2+7x)(9x^2+1/4)\)

FOIL

\(\displaystyle f{}'(x)= 6x^4+2x^2/4+21x^3+7x/4+9x^4+x^2/4+63x^3+7x/4\)

Combine like-terms

\(\displaystyle f{}'(x)= 15x^4 + 84x^3+\frac{3x^2}{4}+\frac{14x}{4}\)

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