Question 1

Suppose the equation for a circle is $$x^2$$ + $$y^2 = 25$. What is the slope of the tangent line at the point (3$$, 4)$$?

Accepted Answer

$$-\frac{3}{4}$$

Question 2

Given $$y = \sin(x^2$)$$, which differentiation rule(s) must be used to compute $$\frac{dy}{dx}$$?

Accepted Answer

Chain rule only

Question 3

If $$y is $$defined implicitly by $$x^2 + xy$$ + $$y^2 = 7$, what is $$\frac{dy}{dx}$$ in terms of x$ and y$?

Accepted Answer

$$\frac{-2x - y}{x + 2y}$$

Question 4

A chemical reaction's rate is modeled by $$R = k \cdot e$$^{-t^2}$$, where k$ is a constant and t$ is time. What is $$\frac{dR}{dt}$$?

Accepted Answer

$$-2kt e$$^{-t^2}$$

Question 5

Given $$y is $$defined implicitly by $$\sin(y) - x y = x$$^{1/3}$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{\frac{1}{3} x$$^{-2/3}$$ + y}{\cos(y) - x}$$

Question 6

Which of the following best describes the chain rule?

Accepted Answer

$$\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$$

Question 7

A medical imaging device models the boundary of an organ as $$x^2/9$$ + $$y^2/4 = 1$. At which points does the tangent line have a vertical slope?

Accepted Answer

$$x = 0$$, $$y = \pm 2$$

Question 8

If $$y is $$defined implicitly by $$e^{4y^2}$$ = $$x^2 + 1$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{x}{4y e$$^{4y^2}$$}$$

Question 9

Which of the following is the correct application of the product rule to $$f(x) = x^2$ \sin(x)$$?

Accepted Answer

$$2x \sin(x) + x^2$ \cos(x)$$

Question 10

Suppose $$y is $$defined implicitly by $$x^2$$ + $$y^2 = 1$. At which points is the tangent line horizontal?

Accepted Answer

$$x = \pm 1$$, $$y = 0$$

Question 11

A population grows according to $$P = A e$$^{kt}$$, where A$ and k$ are constants. What is $$\frac{dP}{dt}$$?

Accepted Answer

$$k A e$$^{kt}$$

Question 12

Which of the following is the correct quotient rule for $$f(x) = \frac{x^2$}{\sin(x)}$$?

Accepted Answer

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

Question 13

If $$y is $$defined implicitly by $$x^2 y$$ + $$y^3 = 7$, what is $$\frac{dy}{dx}$$?

Accepted Answer

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

Question 14

Which of the following best describes implicit differentiation?

Accepted Answer

Differentiating both sides of an equation with respect to $$x$$, treating $$y as a $$function of $$x$$

Question 15

A satellite's path is modeled by $$x^2$$ + $$4y^2 = 16$. What is the slope of the tangent line at the point (2$$, 2)$$?

Accepted Answer

$$-\frac{1}{2}$$

Question 16

If $$y = (3x^2$ + 2)^5$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$30x (3x^2$ + 2)^4$

Question 17

Which of the following is NOT a correct differentiation rule?

Accepted Answer

$$\frac{d}{dx}[u + v] = u'v'$$

Differentiation as an Operator: Chain Rule, Implicit Differentiation, and Applications

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