In each part, find each limit by interpreting the expressionas an appropriate

(a) Find dy/dx by differentiating implicitly. (b) Solve the equation for y as a function of x, and find dy/dx from that equation. (c) Confirm that the two results are consistent by expressing the derivative in part (a) as a function of x alone. \(x^{3}+x y-2 x=1\) Equation Transcription: Text Transcription: x^3 +xy-2x=1

(a) Find dy/dx by differentiating implicitly. (b) Solve the equation for y as a function of x, and find dy/dx from that equation. (c) Confirm that the two results are consistent by expressing the derivative in part (a) as a function of x alone. \(x y=x-y\) Equation Transcription Text Transcription: xy=x-y

Find dy/dx by implicit differentiation. \(\frac{1}{y}+\frac{1}{x}=1\) Equation Transcription: Text Transcription: 1/y +1/x =1

Find dy/dx by implicit differentiation. \(x^{3}-y^{3}=6 x y\) Equation Transcription: Text Transcription: x^3 -y^3 =6xy

Find dy/dx by implicit differentiation. \(\sec (x y)=y\) Equation Transcription: Text Transcription: sec(xy)=y

Find dy/dx by implicit differentiation. \(x^{2}=\frac{\cot y}{1+\csc y}\) Equation Transcription: Text Transcription: x^2 =cot y/1+csc y

Find \(d^{2} y / d x^{2}\) by implicit differentiation. \(3 x^{2}-4 y^{2}=7\) Equation Transcription: Text Transcription: d^2 y/dx^2 3x^2 -4y^2 =7

Find \(d^{2} y / d x^{2}\) by implicit differentiation. \(2 x y-y^{2}=3\) Equation Transcription: Text Transcription: d^2 y/dx^2 2xy-y^2 =3

Use implicit differentiation to find the slope of the tangent line to the curve \(y=x \tan (\pi y / 2), x>0, y>0\) (the quadratrix of Hippias) at the point \(\left(\frac{1}{2}, \frac{1}{2}\right)\). Equation Transcription: Text Transcription: y=x tan(piy/2),x>0,y>0 (1/2 , 1/2)

At what point(s) is the tangent line to the curve \(y^{2}=2 x^{3}\) perpendicular to the line \(4 x-3 y+1=0\) ? Equation Transcription: Text Transcription: y^2 =2x^3 4x-3y+1=0

Prove that if P and Q are two distinct points on the rotated ellipse \(x^{2}+x y+y^{2}=4\) such that P,Q, and the origin are collinear, then the tangent lines to the ellipse at P and Q are parallel. Equation Transcription: Text Transcription: x^2 +xy+y^2 =4

Find the coordinates of the point in the first quadrant at which the tangent line to the curve \(x^{3}-x y+y^{3}=0\) is parallel to the x-axis. Equation Transcription: Text Transcription: x^3 -xy+y^3 =0

Find the coordinates of the point in the first quadrant at which the tangent line to the curve \(x^{3}-x y+y^{3}=0\) is parallel to the y-axis. Equation Transcription: Text Transcription: x^3 -xy+y^3 =0

Use implicit differentiation to show that the equation of the tangent line to the curve \(y^{2}=k x\) at \(\left(x_{0}, y_{0}\right)\) is \(y_{0} y=\frac{1}{2} k\left(x+x_{0}\right)\) Equation Transcription: Text Transcription: y^2 =kx (x_0 ,y_0) y_0 y=½ k(x+x_0)

Find dy/dx by first using algebraic properties of the natural logarithm function. \(y=\ln \left(\frac{(x+1)(x+2)^{2}}{(x+3)^{3}(x+4)^{4}}\right)\) Equation Transcription: Text Transcription: y=ln((x+1)(x+2)^2 /(x+3)^3 (x+4)^4 )

Find dy/dx by first using algebraic properties of the natural logarithm function. \(y=\ln \left(\frac{\sqrt{x} \sqrt[3]{x+1}}{\sin x \sec x}\right)\) Equation Transcription: Text Transcription: y=ln(sqrtx cube root x+1//sin x sec x )

Find \(d y / d x\) \(y=\ln 2 x\) Equation Transcription: Text Transcription: dy/dx y=ln 2x

Find \(d y / d x\) \(y=(\ln x)^{2}\) Equation Transcription: Text Transcription: dy/dx y=(ln x)^2

Find \(d y / d x\) \(y=\sqrt[3]{\ln x+1}\) Equation Transcription: Text Transcription: dy/dx y=3 sqrt ln x + 1

Find \(d y / d x\) \(y=\ln (\sqrt[3]{x+1})\) Equation Transcription: Text Transcription: dy/dx y=ln(3 sqrt x + 1)

Find \(d y / d x\) \(y=\log (\ln x)\) Equation Transcription: Text Transcription: dy/dx y=log(ln x)

Find \(d y / d x\) \(y=\frac{1+\log x}{1-\log x}\) Equation Transcription: Text Transcription: dy/dx y=1 + log x/1 - log x

Find \(d y / d x\) \(y=\ln \left(x^{3 / 2} \sqrt{1+x^{4}}\right)\) Equation Transcription: Text Transcription: dy/dx y=ln(x^3/2 sqrt 1 + x^4)

Find \(d y / d x\) \(y=\ln \left(\frac{\sqrt{x} \cos x}{1+x^{2}}\right)\) Equation Transcription: Text Transcription: dy/dx y=ln (sqrt x cos x/1 + x^2)

Find \(d y / d x\) \(y=e^{\ln \left(x^{2}+1\right)}\) Equation Transcription: Text Transcription: dy/dx y=e^ln(x^2+1)

Find \(d y / d x\) \(y=\ln \left(\frac{1+e^{x}+e^{2 x}}{1-e^{3 x}}\right)\) Equation Transcription: Text Transcription: dy/dx y=ln(1+e^x+e^2x/1-e^3x)

Find \(d y / d x\) \(y=2 x e^{\sqrt{x}}\) Equation Transcription: Text Transcription: dy/dx y=2xe^sqrtx

Find \(d y / d x\) \(y=\frac{a}{1+b e^{-1}}\) Equation Transcription: Text Transcription: dy/dx y=a/a+be^-x

Find \(d y / d x\) \(y=\frac{1}{\pi} \tan ^{-1} 2 x\) Equation Transcription: Text Transcription: dy/dx y=1/pi tan^-1 2x

Find \(d y / d x\) \(y=2^{\sin ^{-1} x}\) Equation Transcription: Text Transcription: dy/dx y=2^sin^-1 x

Find \(d y / d x\) \(y=x^{\left(e^{x}\right)}\) Equation Transcription: Text Transcription: dy/dx y=x^(e^x)

Find \(d y / d x\) \(y=(1+x)^{1 / x}\) Equation Transcription: Text Transcription: dy/dx y=(1+x)1/x

Find \(d y / d x\) \(y=\sec ^{-1}(2 x+1)\) Equation Transcription: Text Transcription: dy/dx y=sec^-1(2x+1)

Find \(d y / d x\) \(y=\sqrt{\cos ^{-1} x^{2}}\) Equation Transcription: Text Transcription: dy/dx y=sqrt cos^-1 x^2

Find \(d y / d x\) using logarithmic differentiation. \(y=\frac{x^{3}}{\sqrt{x^{2}+1}}\) Equation Transcription: Text Transcription: dy/dx y=x^3/sqrt x^2+1

Find \(d y / d x\) using logarithmic differentiation. \(y=\sqrt[3]{\frac{x^{2}-1}{x^{2}+1}}\) Equation Transcription: Text Transcription: dy/dx y=3 sqrt x^2-1/x^2+1

(a) Make a conjecture about the shape of the graph of \(y=\frac{1}{2} x-\ln x\), and draw a rough sketch. (b) Check your conjecture by graphing the equation over the interval \(0<x<5\) with a graphing utility. (c) Show that the slopes of the tangent lines to the curve at \(x=1\) and \(x=e\) have opposite signs. (d) What does part (c) imply about the existence of a horizontal tangent line to the curve? Explain. (e) Find the exact x-coordinates of all horizontal tangent lines to the curve. Equation Transcription: Text Transcription: y=1/2 x?ln x 0<x<5 x=1 x=e

Recall from Section 0.5 that the loudness \(\beta\) of a sound in decibels (dB) is given by \(\beta=10 \log \left(I / I_{0}\right)\), where I is the intensity of the sound in watts per square meter \(\left(W / m^{2}\right)\) and \(I_{0}\) is a constant that is approximately the intensity of a sound at the threshold of human hearing. Find the rate of change of \(\beta\) with respect to I at the point where (a) \(I / I_{0}=10\) (b) \(I / I_{0}=100\) (c) \(I / I_{0}=1000\). Equation Transcription: Text Transcription: beta beta=10 log (I/I_0) (W/m^2) I_0 I/I_0 = 10 I/I_0 = 100 I/I_0 = 1000

A particle is moving along the curve \(y=x \ln x\). Find all values of x at which the rate of change of y with respect to time is three times that of x. [Assume that \(d x / d t\) is never zero.] Equation Transcription: Text Transcription: y=x ln ?x dx/dt

Find the equation of the tangent line to the graph of \(y=\ln \left(5-x^{2}\right)\) at \(x=2\) Equation Transcription: Text Transcription: y=ln(5-x^2) x=2

Find the value of b so that the line \(y=x\) is tangent to the graph of \(y=\log _{b} x\). Equation Transcription: Text Transcription: y=x y=lob_b x

Confirm your result by graphing both \(y=x\) and \(y=\log _{b} x\) in the same coordinate system. In each part, find the value of k for which the graphs of \(y=f(x)\) and \(y=\ln x\) share a common tangent line at their point of intersection. Confirm your result by graphing \(y=f(x)\) and \(y=\ln x\) in the same coordinate system. (a) \(f(x)=\sqrt{x}+k\) (b) \(f(x)=k \sqrt{x}\) Equation Transcription: Text Transcription: y=x y=log_b x y=f(x) y=ln x f(x)=sqrt x+k f(x)=k sqrt x

If \(f\) and \(g\) are inverse functions and \(f\) is differentiable on its domain, must \(g\) be differentiable on its domain? Give a reasonable informal argument to support your answer. Equation Transcription: Text Transcription: f g

In each part, find \(\left(f^{-1}\right)^{\prime}(x)\) using Formula (2) of Section 3.3, and check your answer by differentiating \(f^{-1}\) directly. (a) \(f(x)=3 /(x+1)\) (b) \(f(x)=\sqrt{e^{x}}\) Equation Transcription: Text Transcription: (f^-1)(x) f^-1 f(x) = 3/(x+1) f(x) = sqrt e^x

Find a point on the graph of \(y=e^{3 x}\) at which the tangent line passes through the origin. Equation Transcription: Text Transcription: y=e^3x

Show that the rate of change of \(y=5000 e^{1.07 x}\) is proportional to y. Equation Transcription: Text Transcription: y=5000e^1.07x

Show that the rate of change of \(y=3^{2 x} 5^{7 x}\) is proportional to y. Equation Transcription: Text Transcription: y=3^2x 5^7x

The equilibrium constant k of a balanced chemical reaction changes with the absolute temperature T according to the law \(k=k_{0} \exp \left(-\frac{q\left(T-T_{0}\right)}{2 T_{0} T}\right)\) where \(k_{0}, q\), and \(T_{0}\) are constants. Find the rate of change of k with respect to T. Equation Transcription: Text Transcription: k=k_0 exp(-q(T-T_0)/2T_0 T) k_0, q T_0

Show that the function \(y=e^{a x} \sin b x\) satisfice \(y^{\prime \prime}-2 a y^{\prime}+\left(a^{2}+b^{2}\right) y=0\) for any real constants a and b. Equation Transcription: Text Transcription: y=e^ax sin bx y’’-2ay’ +(a^2+b^2)y=0

Show that the function \(y=\tan ^{-1} x\) satisfies \(y^{\prime \prime}=-2 \sin y \cos ^{3} y\). Equation Transcription: Text Transcription: y=tan^1 x y’’ =-2 sin y cos^3 y

Suppose that the population of deer on an island is modeled by the equation \(P(t)=\frac{95}{5-4 e^{-1 / 4}}\) where \(P(t)\) is the number of deer t weeks after an initial observation at time \(t=0\). (a) Use a graphing utility to graph the function \(P(t)\). (b) In words, explain what happens to the population over time. Check your conclusion by finding \(\lim _{x \rightarrow+\infty} P(t)\) (c) In words, what happens to the rate of population growth over time? Check your conclusion by graphing \(P^{\prime}(t)\) Equation Transcription: Text Transcription: P(t)=95/5-4e^-t/4 P(t) t=0 lim_x rightarrow + inifinity P(t) P'(t)

In each part, find each limit by interpreting the expression as an appropriate derivative. (a) \(\lim _{h \rightarrow 0} \frac{(1+h)^{\pi}-1}{h}\) (b) \(\lim _{x \rightarrow e} \frac{1-\ln x}{(x-e) \ln x}\) Equation Transcription: Text Transcription: lim_h rightarrow 0 (1+h)^pi -1/h lim_x rightarrow e 1-ln x/(x-e)ln x

Suppose that \(\lim f(x)=\pm \infty\) and \(\lim g(x)=\pm \infty\), In each of the four possible cases, state whether \(\lim [f(x)-g(x)]\) is an indeterminate form, and give a reasonable informal argument to support your answer. Equation Transcription: Text Transcription: lim f(x) = +- infinity lim g(x) = +- infinity lim [f(x)-g(x)]

(a) Under what conditions will a limit of the form \(\lim _{x \rightarrow a}[f(x) / g(x)]\) be an indeterminate form? (b) If \(\lim _{x \rightarrow a} g(x)=0\), must \(\lim _{x \rightarrow a}[f(x) / g(x)]\) be an indeterminate form? Give some examples to support your answer. Equation Transcription: Text Transcription: lim_x rightarrow a [f(x)/g(x)] lim_x rightarrow a g(x) = 0 lim_x rightarrow a [f(x)/g(x)]

Evaluate the given limit. \(\lim _{x \rightarrow+\infty}\left(e^{x}-x^{2}\right)\) Equation Transcription: Text Transcription: lim_x rightarrow + infinity (e^x - x^2)

Evaluate the given limit. \(\lim _{x \rightarrow 1} \sqrt{\frac{\ln x}{x^{4}-1}}\) Equation Transcription: Text Transcription: lim_x rightarrow 1 sqrt ln x/x^4 - 1

Evaluate the given limit. \(\lim _{x \rightarrow 0} \frac{x^{2} e^{x}}{\sin ^{2} 3 x}\) Equation Transcription: Text Transcription: lim_x rightarrow 0 x^2 e^x/sin^2 3x

Evaluate the given limit. \(\lim _{x \rightarrow 0} \frac{a^{x}-1}{x}, a>0\) Equation Transcription: Text Transcription: lim_x rightarrow 0 a^x-1/x, a > 0

An oil slick on a lake is surrounded by a floating circular containment boom. As the boom is pulled in, the circular containment area shrinks. If the boom is pulled in at the rate of \(5 \mathrm{~m} / \mathrm{min}\), at what rate is the containment area shrinking when the containment area has a diameter of \(100 \mathrm{~m}\)? Equation Transcription: Text Transcription: 5 m/min 100 m

The hypotenuse of a right triangle is growing at a constant rate of \(a\) centimeters per second and one leg is decreasing at a constant rate of \(b\) centimeters per second. How fast is the acute angle between the hypotenuse and the other leg changing at the instant when both legs are \(1 cm\)? Equation Transcription: Text Transcription:

In each part, use the given information to find \(\Delta x, \Delta y\), and \(d y\). (a) \(y=1 /(x-1) ; x\) decreases from 2 to 1.5. (b) \(y=\tan x ; x\) increases from \(-\pi / 4\) to 0 . (c) \(y=\sqrt{25-x^{2}} ; x\) increases from 0 to 3 . Equation Transcription: Text Transcription: delta x, delta y dy y=1/(x-1);x y=tan x;x -pi/4 y=sqrt 25-x^2 ; x

Use an appropriate local linear approximation to estimate the value of \(\cot 46^{\circ}\), and compare your answer to the value obtained with a calculating device. Equation Transcription: Text Transcription: cot 46 degrees

The base of the Great Pyramid at Giza is a square that is 230 m on each side. (a) As illustrated in the accompanying figure, suppose that an archaeologist standing at the center of a side measures the angle of elevation of the apex to be \(\phi=51^{\circ}\) with an error of \(\pm 0.5^{\circ}\). What can the archaeologist reasonably say about the height of the pyramid? (b) Use differentials to estimate the allowable error in the elevation angle that will ensure that the error in calculating the height is at most \(\pm 5 m\). Equation Transcription: Text Transcription: phi=51 degrees +- 0.5 degrees +- 5 m