Question 1

Which of the following best describes an implicitly defined function?

Accepted Answer

A function where $$x$$ and $$y$$ are related by an equation, but $$y is $$not isolated, such as $$x^2$$ + $$y^2 = 25$.

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

If $$2xy + y^2$ = x + y$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{dy}{dx} = \frac{1 - 2x}{2y + 2x - 1}$$

Question 4

For the equation $$e^{2x}$$ = \sin(x + 3y)$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{dy}{dx} = \frac{2e$$^{2x}$$ - \cos(x + 3y)}{3\cos(x + 3y)}$$

Question 5

Which of the following is the correct derivative of $$y^5$$ with respect to $$x$$?

Accepted Answer

$$5y^4$$ \frac{dy}{dx}$$

Question 6

Given $$x^2 + xy$$ - $$y^2 = 1$ at the point (2$$, 3)$$, what is the slope of the tangent line?

Accepted Answer

$$\frac{5}{7}$$

Question 7

Which of the following best describes the process of implicit differentiation?

Accepted Answer

Differentiate both sides with respect to $$x$$, treating $$y as a $$function of $$x.$$

Question 8

If $$\cos y is $$differentiated with respect to $$x$$, what is the result?

Accepted Answer

$$-\sin y \frac{dy}{dx}$$

Question 9

A topographic map uses contour lines to represent elevation. If the contour lines are close together, what does this indicate about the slope?

Accepted Answer

The slope is steep.

Question 10

Given $$2xy + \pi \sin y = 11\pi at $$the point $$(2, \frac{5\pi}{2})$$, what is the equation of the tangent line?

Accepted Answer

$$y - \frac{5\pi}{2} = m(x - 2)$$, where $$m = -\frac{2}{\pi \cos(\frac{5\pi}{2})}$$

Question 11

Which of the following is the correct derivative of $$\cos y$$ with respect to $$y$$?

Accepted Answer

$$-\sin y$$

Question 12

If $$x^3 = 2x - yx + 3y$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{3x^2$ - 2}{-x + 3}$$

Question 13

Which of the following real-world scenarios best illustrates the use of implicit differentiation?

Accepted Answer

Finding the rate at which the radius of a circular oil spill increases as the area expands.

Question 14

If $$y is a $$function of $$x$$, what is $$\frac{d}{dx}[y^5$]$$?

Accepted Answer

$$5y^4$$ \frac{dy}{dx}$$

Question 15

Which of the following is NOT a step in implicit differentiation?

Accepted Answer

Integrate both sides with respect to $$x.$$

Implicit Differentiation and Implicitly Defined Functions

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