Q: Which of the following is the correct first step in logarithmic differentiation for $y = \dfrac{x^3 (\tan x)^9}{x^2 + e^x}$?

$\ln y = \ln(x^3) + \ln((\tan x)^9) - \ln(x^2 + e^x)$

Q: Which of the following is the correct derivative of $y = x^3 (\tan x)^9 / (x^2 + e^x)$ using logarithmic differentiation?

$y' = y \left[ \dfrac{3}{x} + \dfrac{9 \sec^2 x}{\tan x} - \dfrac{2x + e^x}{x^2 + e^x} \right]$

Question 1

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

Accepted Answer

$-2$

Question 2

For the curve $y^2 = x^2(2-x)$ at $(1,1)$, what is the equation of the normal line?

Accepted Answer

$y = -x + 2$

Question 3

Which of the following is the correct logarithmic property for $\ln(a^b)$?

Accepted Answer

$\ln(a^b) = b \ln a$

Question 4

If $y = x^x$, what is $\dfrac{dy}{dx}$?

Accepted Answer

$x^x(1 + \ln x)$

Question 5

Given $y = (\sin x)^x$, what is $\dfrac{dy}{dx}$?

Accepted Answer

$(\sin x)^x \left( \ln(\sin x) + x \cot x \right)$

Question 6

Which of the following is the correct first step in logarithmic differentiation for $y = \dfrac{x^3 (	an x)^9}{x^2 + e^x}$?

Accepted Answer

$\ln y = \ln(x^3) + \ln((	an x)^9) - \ln(x^2 + e^x)$

Question 7

For the curve $x^2 y^2 = (y+1)^2 (4-y^2)$ at the point $(0,-2)$, what is the slope of the tangent line?

Accepted Answer

$0$

Question 8

If the slope of the tangent line to a curve at a point is $0$, what is the slope of the normal line at that point?

Accepted Answer

$\infty$

Question 9

Which of the following is NOT a correct property of logarithms?

Accepted Answer

$\ln(a + b) = \ln a + \ln b$

Question 10

A biologist models the growth of a bacteria population by $y = x^x$. If $x$ is time in hours, what is the rate of change of the population at $x=2$ hours?

Accepted Answer

$4(1 + \ln 2)$

Question 11

Given $y = \dfrac{(x^2+1)^2 e^{2x}}{\sin^5(3x)}$, which of the following is the correct expression for $\ln y$ to start logarithmic differentiation?

Accepted Answer

$\ln y = 2 \ln(x^2+1) + 2x - 5 \ln(\sin(3x))$

Question 12

A chemical reaction rate is modeled by $y = (\sin x)^x$. What is the instantaneous rate of change of $y$ with respect to $x$ at $x = \dfrac{\pi}{2}$?

Accepted Answer

$1$

Question 13

Which of the following is the correct derivative of $y = x^3 (	an x)^9 / (x^2 + e^x)$ using logarithmic differentiation?

Accepted Answer

$y' = y \left[ \dfrac{3}{x} + \dfrac{9 \sec^2 x}{	an x} - \dfrac{2x + e^x}{x^2 + e^x} ight]$

Question 14

A medical researcher uses the function $y = (x^2+1)^2 e^{2x} / \sin^5(3x)$ to model the concentration of a drug. What is the derivative $y'$ in terms of $y$ and $x$?

Accepted Answer

$y' = y \left[ \dfrac{4x}{x^2+1} + 2 - \dfrac{15 \cos(3x)}{\sin(3x)} \right]$

Question 15

Which of the following is the correct application of the chain rule in logarithmic differentiation for $y = f(x)^{g(x)}$?

Accepted Answer

$\ln y = g(x) \ln f(x)$,\ then\ \dfrac{y'}{y} = g'(x) \ln f(x) + g(x) \dfrac{f'(x)}{f(x)}$

Question 16

A physicist models the displacement of a particle as $y = x^x$. If the displacement at $x=1$ is $1$, what is the instantaneous velocity at $x=1$?

Accepted Answer

$1$

Question 17

Which of the following is the correct logarithmic property for $\ln\left(\dfrac{a}{b}\right)$?

Accepted Answer

$\ln a - \ln b$

Question 18

If $y = (x^2+1)^2 e^{2x} / \sin^5(3x)$, what is $y'$ at $x=0$?

Accepted Answer

$2 e^0 / \sin^5(0)$ (undefined)

Tangents, Normals, and Logarithmic Differentiation in Calculus

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