Exploring Behaviors of Implicit Relations - AP Calculus AB Study Guide

The Core Idea: Exploring Behaviors of Implicit Relations

This topic extends the concept of implicit differentiation to the second derivative. While the first derivative, $\frac{dy}{dx}$, of an implicit relation gives us the slope of the tangent line at any point on the curve, the second derivative, $\frac{d^{2}y}{d{x}^2}$, reveals the concavity of the curve. Finding the second derivative involves differentiating the first derivative expression with respect to $x$.

The key procedure is a two-step differentiation process. After finding $\frac{dy}{dx}$, we differentiate it again. This second step will almost always generate a new $\frac{dy}{dx}$ term, which must be substituted with the expression found in the first step. The final result allows us to analyze the concavity (whether the curve is bending upwards or downwards) at any specific point $(x,y)$ on the implicitly defined curve.

Key Process: Finding the Second Derivative Implicitly

The process for finding $\frac{d^{2}y}{d{x}^2}$ for an implicit relation is a direct application of implicit differentiation, performed twice.

1. Find the First Derivative $(\frac{dy}{dx})$: Differentiate the original implicit equation with respect to $x$, remembering to apply the chain rule to any term involving $y$. Then, algebraically solve for $\frac{dy}{dx}$.

2. Find the Second Derivative $(\frac{d^{2}y}{d{x}^2})$: Differentiate the expression for $\frac{dy}{dx}$ from Step 1 with respect to $x$. This will often require the Quotient Rule or Product Rule. During this step, any differentiation of a $y$ term will again produce a $\frac{dy}{dx}$ factor via the chain rule.

3. Substitute $\frac{dy}{dx}$: Replace the $\frac{dy}{dx}$ term that appears in the second derivative expression with the expression you found in Step 1. This ensures the final expression for $\frac{d^{2}y}{d{x}^2}$ is in terms of only $x$ and $y$.

4. (Optional but Recommended) Simplify: Simplify the resulting expression. It is often possible to use the original implicit equation to simplify the final expression for $\frac{d^{2}y}{d{x}^2}$ further.

Understanding Concavity in Implicit Relations

The primary purpose of finding the second derivative of an implicit relation is to determine the concavity of its graph at a specific point. The second derivative, $\frac{d^{2}y}{d{x}^2}$, measures the rate of change of the slope of the tangent line.

• If $\frac{d^{2}y}{d{x}^2}>0$ at a point $(x,y)$, the slope of the tangent line is increasing. This means the graph of the relation is concave up at that point.

• If $\frac{d^{2}y}{d{x}^2}<0$ at a point $(x,y)$, the slope of the tangent line is decreasing. This means the graph of the relation is concave down at that point.

The crucial step in this analysis is substituting the expression for $\frac{dy}{dx}$ back into the expression for $\frac{d^{2}y}{d{x}^2}$. Without this substitution, you cannot evaluate the second derivative using only the coordinates of a point $(x,y)$. The final expression for $\frac{d^{2}y}{d{x}^2}$ must be a function of $x$ and $y$ alone to be used for determining concavity at a point on the curve.

Core Concepts & Rules

• The second derivative of an implicit relation, $\frac{d^{2}y}{d{x}^2}$, is calculated by differentiating the first derivative, $\frac{dy}{dx}$, with respect to $x$.

• The process of finding $\frac{d^{2}y}{d{x}^2}$ requires applying implicit differentiation twice.

• When differentiating the expression for $\frac{dy}{dx}$, the chain rule will introduce a new $\frac{dy}{dx}$ term that must be handled.

• To obtain a final expression for $\frac{d^{2}y}{d{x}^2}$ in terms of only $x$ and $y$, the expression for $\frac{dy}{dx}$ must be substituted back into the second derivative.

• The sign of $\frac{d^{2}y}{d{x}^2}$ evaluated at a point $(x,y)$` determines the concavity of the curve at that point. A positive value indicates the curve is concave up, and a negative value indicates the curve is concave down.

Step-by-Step Example 1: Finding the Second Derivative

Problem: For the curve defined by $y^2-x^2=16$, find an expression for $\frac{d^{2}y}{d{x}^2}$ in terms of $x$ and $y$.

Step 1: Find the first derivative, $\frac{dy}{dx}$

Differentiate the entire equation with respect to $x$:

$$\frac{d}{dx}(y^2-x^2)=\frac{d}{dx}(16)$$

$$2y\frac{dy}{dx}-2x=0$$

Now, solve for $\frac{dy}{dx}$:

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

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

Step 2: Differentiate $\frac{dy}{dx}$ to find $\frac{d^{2}y}{d{x}^2}$

We need to differentiate $\frac{dy}{dx}=\frac{x}{y}$ with respect to $x$. This requires the Quotient Rule: $\frac{g{f}^{\prime }-f{g}^{\prime }}{g^2}$, where $f(x)=x$ and $g(x)=y$.

$$\frac{d^{2}y}{d{x}^2}=\frac{d}{dx}\left(\frac{x}{y}\right)=\frac{(y)(\frac{d}{dx}(x))-(x)(\frac{d}{dx}(y))}{y^2}$$

$$\frac{d^{2}y}{d{x}^2}=\frac{(y)(1)-(x)(\frac{dy}{dx})}{y^2}=\frac{y-x\frac{dy}{dx}}{y^2}$$

Step 3: Substitute the expression for $\frac{dy}{dx}$

Substitute $\frac{dy}{dx}=\frac{x}{y}$ into the expression for $\frac{d^{2}y}{d{x}^2}$:

$$\frac{d^{2}y}{d{x}^2}=\frac{y-x\left(\frac{x}{y}\right)}{y^2}=\frac{y-\frac{x^2}{y}}{y^2}$$

Step 4: Simplify the expression

To simplify the complex fraction, multiply the numerator and denominator by $y$:

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

Now, use the original equation, $y^2-x^2=16$, to simplify the numerator:

$$\frac{d^{2}y}{d{x}^2}=\frac{16}{y^3}$$

Step-by-Step Example 2: Exam-Style Application

Problem: Consider the curve given by the equation $2y^3+6x^{2}y-12x^2+6y=1$.

(a) Find $\frac{dy}{dx}$.

(b) Determine the concavity of the curve at the point $(1,-1)$.

(a) Find $\frac{dy}{dx}$

Differentiate the equation with respect to $x$. The term $6x^{2}y$ requires the product rule.

$$\frac{d}{dx}(2y^3+6x^{2}y-12x^2+6y)=\frac{d}{dx}(1)$$

$$6y^2\frac{dy}{dx}+\left[(12x)(y)+(6x^2)(\frac{dy}{dx})\right]-24x+6\frac{dy}{dx}=0$$

Group the terms with $\frac{dy}{dx}$ on one side and all other terms on the other side.

$$6y^2\frac{dy}{dx}+6x^2\frac{dy}{dx}+6\frac{dy}{dx}=24x-12xy$$

Factor out $\frac{dy}{dx}$:

$$\frac{dy}{dx}(6y^2+6x^2+6)=24x-12xy$$

Solve for $\frac{dy}{dx}$:

$$\frac{dy}{dx}=\frac{24x-12xy}{6y^2+6x^2+6}=\frac{4x-2xy}{y^2+x^2+1}$$

(b) Determine the concavity of the curve at the point $(1,-1)$

To determine concavity, we need to find the sign of $\frac{d^{2}y}{d{x}^2}$ at $(1,-1)$. First, let's find the value of $\frac{dy}{dx}$ at this point, as we will need it for the second derivative calculation.

$$\frac{dy}{dx}{|}_{(1,-1)}=\frac{4(1)-2(1)(-1)}{(-1{)}^2+(1{)}^2+1}=\frac{4+2}{1+1+1}=\frac{6}{3}=2$$

Now, we must find $\frac{d^{2}y}{d{x}^2}$. It is often easier to differentiate the unfactored form from part (a) before solving for $\frac{dy}{dx}$. Let's differentiate $6y^2\frac{dy}{dx}+6x^2\frac{dy}{dx}+6\frac{dy}{dx}=24x-12xy$.

$$\frac{d}{dx}\left((6y^2+6x^2+6)\frac{dy}{dx}\right)=\frac{d}{dx}(24x-12xy)$$

Using the product rule on both sides:

$$\left(12y\frac{dy}{dx}+12x\right)\frac{dy}{dx}+(6y^2+6x^2+6)\frac{d^{2}y}{d{x}^2}=24-\left(12y+12x\frac{dy}{dx}\right)$$

We do not need to solve for $\frac{d^{2}y}{d{x}^2}$ algebraically. We can substitute the known values for $x$, $y$, and $\frac{dy}{dx}$ at the point $(1,-1)$.

Substitute $x=1$, $y=-1$, and $\frac{dy}{dx}=2$:

$$(12(-1)(2)+12(1))(2)+(6(-1{)}^2+6(1{)}^2+6)\frac{d^{2}y}{d{x}^2}=24-(12(-1)+12(1)(2))$$

$$(-24+12)(2)+(6+6+6)\frac{d^{2}y}{d{x}^2}=24-(-12+24)$$

$$(-12)(2)+(18)\frac{d^{2}y}{d{x}^2}=24-(12)$$

$$-24+18\frac{d^{2}y}{d{x}^2}=12$$

$$18\frac{d^{2}y}{d{x}^2}=36$$

$$\frac{d^{2}y}{d{x}^2}=\frac{36}{18}=2$$

Since $\frac{d^{2}y}{d{x}^2}=2>0$ at the point $(1,-1)$, the curve is concave up at this point.

Using Your Calculator

This topic is almost exclusively analytical. A calculator cannot be used to find the symbolic second derivative of an implicit relation. Its primary use is for arithmetic calculations after the calculus is complete.

For example, in a problem that asks for the concavity at a point $(a,b)$, you would perform all the differentiation steps by hand to find an expression for $\frac{d^{2}y}{d{x}^2}$. Then, you would substitute $x=a$ and $y=b$ into the expression. A calculator can be useful at this final stage to compute the numerical value and determine its sign, especially if the expression is complex.

• Step 1: Find $\frac{d^{2}y}{d{x}^2}$ in terms of $x$ and $y$ by hand.

• Step 2: Substitute the coordinates of the given point into your expression.

• Step 3: Use the calculator to evaluate the resulting numerical expression to avoid arithmetic errors.

AP Exam Quick Hit

Common Question Types

• Finding the Expression for $\frac{d^{2}y}{d{x}^2}$: A multiple-choice or free-response question might ask you to find the second derivative of an implicit relation, like $x^2+xy+y^2=3$, and express the answer in terms of $x$ and $y$.

• Determining Concavity at a Point: Given an implicit relation and a point on its curve, you will be asked to find the value of $\frac{d^{2}y}{d{x}^2}$ at that point and use its sign to determine if the curve is concave up or concave down.

• Applying the Second Derivative Test: You might be asked to find horizontal tangents (where $\frac{dy}{dx}=0$) and then use the sign of $\frac{d^{2}y}{d{x}^2}$ at those points to classify them as local maxima ($\frac{d^{2}y}{d{x}^2}<0$) or local minima ($\frac{d^{2}y}{d{x}^2}>0$).

Common Mistakes

• Forgetting to Substitute for $\frac{dy}{dx}$: The most common mistake is correctly finding an expression for $\frac{d^{2}y}{d{x}^2}$$ but leaving it in terms of $x$, $y$, and $\frac{dy}{dx}$. You must substitute the expression for $\frac{dy}{dx}$ back in to get full credit.

• Quotient/Product Rule Errors: Making an algebraic mistake while applying the quotient or product rule to the expression for $\frac{dy}{dx}$. Be methodical and write out each step.

• Chain Rule Errors: Forgetting to multiply by $\frac{dy}{dx}$ when differentiating a $y$ term. This error can happen in both the first and second differentiation steps.

• Simplification Errors: Making an error when simplifying the complex fraction that often results from the substitution step.

• Ignoring the Original Equation: Failing to use the original implicit relation to simplify the final expression for $\frac{d^{2}y}{d{x}^2}$. This can leave you with a correct but overly complicated answer.