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

스터디 가이드 - 연습 문제

학습 가이드 - 플래시카드