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Calculus 1 : Functions

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10 Diagnostic Tests 438 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #273 : Equations

Differentiate the logarithm.

$\ln(4x)$

Possible Answers:

$\frac{1}{x}$

$4\ln(4x)$

$\frac{1}{4x}$

Correct answer:

$\frac{1}{x}$

Explanation:

Using the chain rule, we will determine the derivative of our function will be $\frac{d\log(u)}{du}\cdot \frac{du}{dx}$ .

The derivative of the log function is $\frac{1}{x}$ , and our second term of the chain rule will cancel out $\frac{1}{4x}\cdot 4=\frac{4}{4x}=\frac{1}{x}$ .

Thus our derivative will be $\frac{1}{x}$ .

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Example Question #32 : Differential Equations

Differentiate the polynomial.

$x^{4}+3x^{3}+4x$

Possible Answers:

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

$4x^{3}+9x^{2}+4$

$4x^{3}+9x^{2}$

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

x^2+4x+7

Correct answer:

$4x^{3}+9x^{2}+4$

Explanation:

Using the power rule, we can differentiate our first term reducing the power by one and multiplying our term by the original power. $x^{4}$ , will thus become $4x^{3}$ . The second term $3x^{3}$ , will thus become $9x^{2}$ . The last term is , will reduce to .

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Example Question #37 : Differential Equations

Find {f}'(x) .

$f(x) = \frac{e^x}{x^2}$

Possible Answers:

$\frac{x^2-e^x}{x^2}$

$\frac{e^x (x-2)}{x^3}$

$\frac{e^{x^{2}} (x-2)}{x^3}$

$\frac{(x-2)}{x^2}$

$\frac{(x-2)}{x^3}$

Correct answer:

$\frac{e^x (x-2)}{x^3}$

Explanation:

According to the quotient rule, the derivative of ,

$\frac{f(x)}{g(x)} = \frac{{f}'(x)g(x) - f(x){g}'(x))}{(g(x))^{2}}$ .

We will let $f(x) = e^{x} , f{}'(x) = e^{x} , g(x) = x^{2}$ and g{}'(x) = 2x
Plugging all of our values into the quotient rule formula we come to a final solution of :

$\frac{e^x (x-2)}{x^3}$ .

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Example Question #281 : Equations

Differentiate the polynomial.

$x^{3}+3x^{2}$

Possible Answers:

$3x^{2}+6x$

$3x^{4}+6x^{3}$

$x^{3}+3$

$3x^{3}+6x$

Correct answer:

$3x^{2}+6x$

Explanation:

Using the power rule, we can differentiate our first term reducing the power by one and multiplying our term by the original power. $x^{3}$ , will thus become $3x^{2}$ . The second term $3x^{2}$ , will thus become .

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Example Question #33 : Differential Equations

Solve the differential equation: $\frac{dy}{dx}= \frac{y}{2}$

Possible Answers:

y=Ce^0^.^5^x

y=Ce^ -^0^.^5^x

y=Ce^2^x

y=Ce^ -^0^.^2^5^x

y=Ce^x

Correct answer:

y=Ce^0^.^5^x

Explanation:

Rewrite $\frac{dy}{dx}= \frac{y}{2}$ by multiply the on both sides, and dividing on both sides of the equation.

$\frac{1}{y}dy= \frac{1}{2}dx$

Integrate both sides of the equation and solve for y.

$\int\frac{1}{y}dy= \int\frac{1}{2}dx$

$ln\left | y\right |=\frac{1}{2}x+C$

$\left | y\right |= e^0^.^5^xe^C$

y=Ce^0^.^5^x

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Example Question #31 : How To Find Solutions To Differential Equations

Find the equilibrium values for the following differential equation: y^2 + y'=y

Possible Answers:

0,1

-1,1

Correct answer:

0,1

Explanation:

To find the equilibrium values, substitute y'=0 and solve for . The equilibrium values are the solutions of the differential equation in constant form.

y^2 + y'=y

y^2 + 0=y

y^2-y=0

y(y-1)=0

y=0,1

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Example Question #42 : Differential Equations

Suppose is greater than zero. Solve the differential equation: $y'= \frac{x^2}{y^\frac{1}{2}}$

Possible Answers:

$y=\left(\frac{1}{2}x^3+C\right)^\frac{2}{3}$

$y=(2x^3+C)^\frac{2}{3}$

$y=\left(\frac{1}{2}x^3+C\right)^\frac{3}{2}$

$y=\left(\frac{3}{2}x^3+C\right)^\frac{2}{3}$

$y=\left(\frac{2}{3}x^3+C\right)^\frac{2}{3}$

Correct answer:

$y=\left(\frac{1}{2}x^3+C\right)^\frac{2}{3}$

Explanation:

Rewrite $y'= \frac{x^2}{y^\frac{1}{2}}$ so that the same variables $\textup{x, dx, y, dy}$ are aligned correctly on the left and right of the equal sign.

$y^\frac{1}{2}\cdot y'= x^2$

$y^\frac{1}{2}\cdot \frac{dy}{dx}= x^2$

$y^\frac{1}{2}\cdot {dy}= x^2 \cdot dx$

Integrate both sides and solve for .

$\int y^\frac{1}{2}\cdot {dy}= \int x^2 \cdot dx$

$\frac{2}{3}y^\frac{3}{2}=\frac{1}{3}x^3+C$

$y^\frac{3}{2}=\left(\frac{3}{2}\right)\left(\frac{1}{3}x^3+C\right)$

$y^\frac{3}{2}=\frac{1}{2}x^3+\frac{3}{2}C$

$y=\left(\frac{1}{2}x^3+C\right)^\frac{2}{3}$

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Example Question #281 : Equations

Find the function whose slope at the point (x,y) is -4x^3+5x and passes through the point (4,5) .

Possible Answers:

$y= -x^4+\frac{5}{2}x^2+c$

y=181

y=0

$y= -x^4+\frac{5}{2}x^2$

$y= -x^4+\frac{5}{2}x^2+221$

Correct answer:

$y= -x^4+\frac{5}{2}x^2+221$

Explanation:

This question requires us to use differential equations. Begin as follows:

$\frac{dy}{dx}=-4x^3+5x$

dy}=-4x^3+5xdx

$y=\int -4x^3+5xdx$

$y= -x^4+\frac{5}{2}x^2+c$

Now, we know that y must pass through the point (4,5), so we can use this point to find c

$5= -(4)^4+\frac{5}{2}4^2+c$

5= -256+40+c

c=221

So our function is as follows:

$y= -x^4+\frac{5}{2}x^2+221$

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Example Question #281 : Equations

Solve:

$\frac{dy}{dx}=10y$

Possible Answers:

y=Ce^0^.^1^x

$y=\frac{1}{10}e^x$

y=Ce^x

y=10e^x

y=Ce^1^0^x

Correct answer:

y=Ce^1^0^x

Explanation:

Multiply and divide on both sides of the equation $\frac{dy}{dx}=10y$ .

$\frac{1}{y}dy=10 \: dx$

Integrate both sides.

$\int \frac{1}{y}dy=\int 10\: dx$

$ln\left |y \right |=10x+C$

Use base to eliminate the natural log.

$e^{ln(y)}=e^1^0^x^+^C$

y=Ce^1^0^x

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Example Question #41 : Differential Equations

$y=e^{2x}$ .

Calculate y'(1)-y'(0).

Possible Answers:

10.2

12.78

7.39

Correct answer:

12.78

Explanation:

Remember that the derivative of $e^u = e^u \cdot u'$ .

$y'=2e^{2x}$ .

Now plug in the x=0,1 to find the corresponding values.

y'(1)=14.78

y'(0)=2.

Substitute these into the desired formula.

y'(1)-y'(0).

14.78-2=12.78

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Calculus 1 : Functions

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #273 : Equations

Example Question #32 : Differential Equations

Example Question #37 : Differential Equations

Example Question #281 : Equations

Example Question #33 : Differential Equations

Example Question #31 : How To Find Solutions To Differential Equations

Example Question #42 : Differential Equations

Example Question #281 : Equations

Example Question #281 : Equations

Example Question #41 : Differential Equations

All Calculus 1 Resources

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