Question 1

Given the function $$y^2$$ + $$x^2 = 1$, use implicit differentiation to find $$\frac{dy}{dx}$$ in terms of x$ and y$.

Accepted Answer

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

Question 2

Suppose $$y is $$defined implicitly by $$x^2$$ + $$y^2 = 1$. What is $$\frac{dy}{dx}$$ at the point (0$$,1)$$?

Accepted Answer

$$\frac{dy}{dx} = 0$$

Question 3

Given $$y = \cos x$$, use implicit differentiation to find $$\frac{dy}{dx}.$$

Accepted Answer

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

Question 4

The function $$y = \arccos x is $$the inverse cosine function. What is $$\frac{d}{dx}(\arccos x)$$?

Accepted Answer

$$\frac{d}{dx}(\arccos x) = -\frac{1}{\sqrt{1-x^2$}}$$

Question 5

Suppose $$y = \arctan x. $$What is $$\frac{d}{dx}(\arctan x)$$?

Accepted Answer

$$\frac{d}{dx}(\arctan x) = \frac{1}{1+x^2$}$$

Question 6

A small paper cup is being filled with water. The cup is shaped like a cone with height $$h$$ and radius $$r at $$the top. If the water is filled to a height $$y$$, what is the radius of the water surface at that height?

Accepted Answer

$$r_{water} = \frac{r}{h}y$$

Question 7

If the volume of water in a conical cup is $$V = \frac{1}{3}\pi r^2$ y$$ and $$r = \frac{R}{H}y$$, express $$V in $$terms of $$y$$, $$R$$, and $$H$$ only.

Accepted Answer

$$V = \frac{1}{3}\pi \left(\frac{R}{H}y\$$right)^2$$ y = \frac{1}{3}\pi \frac{R^2$}{H^2$} y^3$

Question 8

Given $$V = \frac{1}{3}\pi \frac{R^2$}{H^2$} y^3$, find $$\frac{dV}{dy}$$.

Accepted Answer

$$\frac{dV}{dy} = \pi \frac{R^2$}{H^2$} y^2$

Question 9

If water is poured into a conical cup at a rate of $$\frac{dV}{dt} = 2 cm$^3/s$$, and when $$y = 2 cm$$, $$R = 5 cm$$, $$H = 10 cm$$, what is $$\frac{dy}{dt} at $$that instant?

Accepted Answer

$$\frac{dy}{dt} = \frac{2}{\pi \frac{25}{100} \cdot 4} = \frac{2}{\pi} cm/s$$

Question 10

If $$f$$ and $$f^{-1}$$ are inverse functions, which formula correctly relates their derivatives?

Accepted Answer

$$(f^{-1}$$)'(y) = \frac{1}{f'(x)}$$ where x$$ = $$f^{-1}(y$$)$$

Question 11

Given the graph of $$y = f(x)$$, how can you use the tangent lines to estimate $$(f^{-1}$$)'(y)$$ at a point?

Accepted Answer

Find the slope of the tangent line to $$f at$$ $$x$$, then take its reciprocal.

Question 12

If $$y = \sin x$$, use implicit differentiation to find $$\frac{dx}{dy}.$$

Accepted Answer

$$\frac{dx}{dy} = \frac{1}{\cos x}$$

Question 13

Suppose $$y = \arcsin x. $$What is $$\frac{d}{dx}(\arcsin x)$$?

Accepted Answer

$$\frac{d}{dx}(\arcsin x) = \frac{1}{\sqrt{1-x^2$}}$$

Question 14

Given $$x^2$$ + $$y^2 = 1$, what is the equation of the tangent line to the curve at the point $$(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}})$$?

Accepted Answer

$$y - \frac{1}{\sqrt{2}} = -1 \left(x - \frac{1}{\sqrt{2}}\right)$$

Question 15

If $$y^2$$ + $$x^2 = 1$ and y$$ \u003e 0$$, what is $$\frac{dy}{dx}$$ at x$$ = \frac{1}{2}$$?

Accepted Answer

$$\frac{dy}{dx} = -\frac{1}{2\sqrt{3}/2} = -\frac{1}{\sqrt{3}}$$

Question 16

Suppose the rate of change of water height in a conical cup is $$\frac{dy}{dt}$$ and the rate of change of volume is $$\frac{dV}{dt}. $$Which formula correctly relates $$\frac{dy}{dt}$$ and $$\frac{dV}{dt}$$?

Accepted Answer

$$\frac{dy}{dt} = \frac{1}{\frac{dV}{dy}} \frac{dV}{dt}$$

Question 17

If $$y = \arctan x$$, use implicit differentiation to find $$\frac{dx}{dy}.$$

Accepted Answer

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

Implicit Differentiation, Inverse Functions, and Related Rates: Calculus Lab Study Guide

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