Q: For the function $$y = \ln \left( \frac{e$$^{-x}$$ \cos^2$ x}{x^2$ + x + 1} \right)$$, what is $$y'$$?

$$y' = \frac{2x + 1}{x^2$ + x + 1} - 2 \tan x - 1$$

Question 1

Given $$y = \log_2$(e^{-x}$$ \cos(nx))$$, which of the following is the correct expression for y$$'$?

Accepted Answer

$$y' = \frac{-e$$^{-x}$$ \cos(nx) + e$$^{-x}$$ \sin(nx) \cdot n}{e$$^{-x}$$ \cos(nx) \ln 2}$$

Question 2

For the function $$y = \frac{\ln x}{x^2$}$$, what is the second derivative $$y''$$?

Accepted Answer

$$y'' = \frac{x^2$(1 - 2\ln x + 2) - (x - 2\ln x) \cdot 4x}{x^4$}$$

Question 3

Suppose $$y = (\sin x$$)^{\ln x}$$. What is y$$'$$?

Accepted Answer

$$y' = \frac{\cot x}{x} (\sin x$$)^{\ln x}$$

Question 4

Given $$y = \ln \left( \frac{e$$^{-x}$$ \cos^2$ x}{x^2$ + x + 1} \right)$$, which of the following is the correct simplification for $$y$$?

Accepted Answer

$$y = -x + 2 \ln(\cos x) - \ln(x^2$ + x + 1)$$

Question 5

For the function $$y = \ln \left( \frac{e$$^{-x}$$ \cos^2$ x}{x^2$ + x + 1} \right)$$, what is $$y'$$?

Accepted Answer

$$y' = \frac{2x + 1}{x^2$ + x + 1} - 2 	an x - 1$$

Question 6

A pharmaceutical company models the concentration of a drug in the bloodstream as $$C(t) = \frac{\ln t}{t^2$}$$, where $$t is $$time in hours. What does the sign of $$C'(t)$$ indicate about the drug concentration over time?

Accepted Answer

If $$C'(t) \u003e 0$$, the concentration is increasing; if $$C'(t) \u003c 0$$, it is decreasing.

Question 7

If $$y = \log_2$(e^{-x}$$ \cos(nx))$$, which of the following best describes the domain of y$?

Accepted Answer

$$x$$ such that $$e^{-x}$$ \cos(nx) \u003e 0$$

Question 8

A biologist models population growth with $$P(x) = (\sin x$$)^{\ln x}$$ for x$$ \u003e 0$$. What is the critical point condition for P(x$$)$$?

Accepted Answer

$$\cot x = 0 or$$ $$\ln x = 0$$

Question 9

Given $$y = \frac{\ln x}{x^2$}$$, which of the following is the correct expression for $$y'$$?

Accepted Answer

$$y' = \frac{1 - 2 \ln x}{x^3$}$$

Question 10

A physicist uses $$y = \log_2$(e^{-x}$$ \cos(nx))$$ to model a signal. If n$ is an integer, which value of x$ will cause y$ to be undefined?

Accepted Answer

$$x$$ such that $$\cos(nx) = 0$$

Question 11

For $$y = (\sin x$$)^{\ln x}$$, which of the following is the correct logarithmic form for $$\ln y$$?

Accepted Answer

$$\ln y = \ln x \ln(\sin x)$$

Question 12

Suppose $$y = \ln \left( \frac{e$$^{-x}$$ \cos^2$ x}{x^2$ + x + 1} \right). $$Which of the following is the correct derivative $$y'$$?

Accepted Answer

$$y' = \frac{2x + 1}{x^2$ + x + 1} - 2 	an x - 1$$

Question 13

A chemical reaction rate is modeled by $$R(x) = \frac{\ln x}{x^2$}. $$What is the value of $$R'(1)$$?

Accepted Answer

$$R'(1) = -2$$

Question 14

If $$y = (\sin x$$)^{\ln x}$$, what is the value of y$$'$$ at x$$ = \frac{\pi}{2}$$?

Accepted Answer

$$y' = 0$$

Question 15

A medical researcher models the decay of a substance with $$y = \log_2$(e^{-x}$$ \cos(nx))$$. What is the effect of increasing x$ on y$ if $$\cos(nx)$ remains positive?

Accepted Answer

The value of $$y$$ decreases.

Question 16

For $$y = \frac{\ln x}{x^2$}$$, which of the following best describes the behavior of $$y as$$ $$x 	o \infty$$?

Accepted Answer

$$y 	o 0$$

Question 17

Given $$y = (\sin x$$)^{\ln x}$$, which of the following is a necessary condition for y$ to be real-valued?

Accepted Answer

$$\sin x \u003e 0$$ and $$x \u003e 0$$

Advanced Differentiation Techniques in Calculus

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