Question 1

Which of the following best describes the difference between explicit and implicit equations?

Accepted Answer

Explicit equations state the relationship between variables directly, while implicit equations express the relationship indirectly.

Question 2

Given the equation $x^2 + y^2 = 25$, what is $\frac{dy}{dx}$ using implicit differentiation?

Accepted Answer

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

Question 3

Why is implicit differentiation necessary for some equations?

Accepted Answer

Because some equations cannot be solved for $y$ explicitly in terms of $x$.

Question 4

Given $x^2 + y^2 = 25$, what are the slopes of the tangent lines at the points where $x = 3$?

Accepted Answer

The slopes are $-\frac{3}{4}$ and $\frac{3}{4}$.

Question 5

Suppose $y$ is defined implicitly by $x^3 + y^3 = 6xy$. Find $\frac{dy}{dx}$.

Accepted Answer

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

Question 6

Given the equation $x^2 + xy + y^2 = 7$, find $\frac{dy}{dx}$.

Accepted Answer

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

Question 7

A company’s cost and revenue are related by the equation $C^2 + R^2 = 100$. If $R$ is a function of $C$, what is $\frac{dR}{dC}$?

Accepted Answer

$\frac{dR}{dC} = -\frac{C}{R}$

Question 8

Given $x^2 + y^2 = 25$, what are the equations of the tangent lines at the points where $x = 3$?

Accepted Answer

$y + 4 = -\frac{3}{4}(x - 3)$ and $y - 4 = \frac{3}{4}(x - 3)$

Question 9

If $x$ and $y$ are related by $x^2 + xy + y^2 = 7$, what is the slope of the tangent line at the point $(1, 2)$?

Accepted Answer

$-\frac{4}{5}$

Question 10

Given the equation $x^2 + y^2 = 25$, why are there two tangent lines at $x = 3$?

Accepted Answer

Because for $x = 3$, there are two corresponding $y$ values on the circle.

Question 11

Suppose $x$ and $y$ are related by $x^2 + y^2 = 25$. If $x = 0$, what are the possible slopes of the tangent lines?

Accepted Answer

The slopes are $0$ (horizontal) and undefined (vertical).

Question 12

Given $x^2 + y^2 = 25$, what is the geometric shape represented by this equation?

Accepted Answer

A circle with center at $(0, 0)$ and radius $5$.

Question 13

If $x$ and $y$ are related by $x^2 + y^2 = 25$, and $y$ is a function of $x$, what is the domain of $x$?

Accepted Answer

$-5 \leq x \leq 5$

Question 14

Given $x^2 + y^2 = 25$, what is $\frac{dy}{dx}$ at the point $(0, 5)$?

Accepted Answer

$0$

Question 15

Suppose $x$ and $y$ are related by $x^2 + y^2 = 25$. If $y$ is negative, what is the sign of $\frac{dy}{dx}$ for $x > 0$?

Accepted Answer

Negative

Question 16

Given $x^2 + y^2 = 25$, what is the equation of the tangent line at the point $(0, 5)$?

Accepted Answer

$y = 5$

Implicit Differentiation- Chapter 3

Study Guide - Practice Questions