# Calculus Homework Help

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## Calculus Help Topics We Can Help You

This is a list of subjects you may be interested on

**Functions**- Polynomials
- Rational Functions
- Logarithms
- Exponentials
- Trigonometric

**Limits**- Direct Substitution Property
- Undetermined Forms
- Trigonometric Limits
- Factorization
- Side Limits
- L'Hopital Rule

**Continuity**- Connection with limits
- Algebraic Properties
- Intermediate Value Theorem

**Derivatives**- Tangent Line
- Product Rule
- Quotient Rule
- Chain Rule
- Related Rates
- Mean Value Theorem
- Maximum and Minimum values

**Integrals**- Antiderivatives
- Fundamental Theorem of Calculus
- Substitutions and change of variables
- Partial Fractions
- Improper Integrals

**Series**- Positive Series
- Convergence criteria
- Alternating Series

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#### All Calculus Subjects

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#### Differentiation

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#### Integration

This one can be a toughie for many. Integration may require advanced understanding of several Calculus topics, and may require of a great deal of creativity. We can help you with this too.

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We know how it feels to deal with hard calculus problems. With our help, you'll get that extra edge you need! No need to sweat with poorly explained problems from the books. We can provide solutions that suit your own needs. See some examples:

**Problem 1.** Find

**Solution.**

The problem presents the typical case of a quotient of two quantities that go to infinity. In fact, $2n^2+1$ and $3n^2-5n+2$ both approach to infinity as $n\rightarrow\infty$. Which means that the quotient

\[\displaystyle\frac{2n^2+1}{3n^2-5n+2}\]is an undetermined form. Somehow we have to cancel the "bad" part, if possible. Notice that since both $2n^2+1$ and $3n^2-5n+2$ are big, there's a chance for cancellation, so the quotient could be finite.

The trick in this case is to divide by the highest power on the expression (from both numerator and denominator). In this case we will divide both numerator and denominator by $n^2$. We get

\[ \displaystyle\frac{2n^2+1}{3n^2-5n+2}= \frac{2+\frac{1}{n^2}}{ 3-\frac{5}{n}+\frac{2}{n^2} }\rightarrow \frac{2}{3} \]as $n\rightarrow\infty$, because $\frac{1}{n^2}\rightarrow 0$, $\frac{5}{n}\rightarrow 0$, and $\frac{2}{n^2} \rightarrow 0$, as $n\rightarrow\infty$.

This trick is applied in general for this type of expressions. $\Box$

**Problem 2.** Find $\displaystyle\frac{dy}{dx}$ and $\displaystyle\frac{d^2 y}{dx^2}$ if

**Solution.** Let's differentiate both sides of the equality. Let's recall that we are assuming that $y=y(x)$. That means that the
equation gives us a way to *solve* $y$ as a function of $x$, but we still *don't know* how to express that function. That
assumption requires further justification (meaning, it's not always granted), but we'll take it for granted. So, since $y=y(x)$ we get
using the *chain rule*:

where $y'\equiv \frac{dy}{dx}$. The 0 on the right hand side comes from differentiating the constant $a^{2/3}$. We multiply the equation by $3/2$ and we get

\[ \displaystyle x^{-1/2}+ y^{-1/3} y'=0 \]which means that

\[\displaystyle \frac{dy}{dx} = -\left(\frac{y}{x}\right)^{1/3} \]Let's now get $y''$. From the last equation we obtain easily that

\[ \displaystyle x\left(\frac{dy}{dx}\right)^3=-y \]and now we differentiate this again with respect to $x$ to get:

\[\displaystyle \left(\frac{dy}{dx}\right)^3+3x\left(\frac{dy}{dx}\right)^2 y''=-y' \]But $\left(\frac{dy}{dx}\right)^3=-y/x$ and $3x\left(\frac{dy}{dx}\right)^2=3x(y/x)^{2/3}$. We know put all the elements together:

\[\displaystyle y''=\frac{y/x+(y/x)^{1/3}}{3x(y/x)^{2/3}} \]and now we simplify the right hand side of the last equation

\[\displaystyle y''=\frac{(y/x)^{2/3}+1}{3x(y/x)^{1/3}}=\frac{y^{2/3}+x^{2/3}}{3x^{4/3}y^{1/3}} \]But by definition: $x^{2/3}+y^{2/3}=a^{2/3}$, so we replace this into the previous equation:

\[\displaystyle y''=\frac{a^{2/3}}{3x^{4/3}y^{1/3}} \]which gives the final answer. $\Box$