?Orthogonal Trajectories Two curves are orthogonal if their tangent lines are

(a) Find \(y^{\prime}\) by implicit differentiation. (b) Solve the equation explicitly for \(y\) and differentiate to get \(y^{\prime}\) in terms of \(x\). (c) Check that your solutions to parts (a) and (b) are consistent by substituting the expression for \(y\) into your solution for part (a). \(5 x^{2}-y^{3}=7\) ________________ Equation Transcription: Text Transcription: y’ y y’ x y 5x^2 - y^3 = 7

(a) Find \(y^{\prime}\) by implicit differentiation. (b) Solve the equation explicitly for \(y\) and differentiate to get \(y^{\prime}\) in terms of \(x\). (c) Check that your solutions to parts (a) and (b) are consistent by substituting the expression for \(y\) into your solution for part (a). \(6 x^{4}+y^{5}=2 x\) ________________ Equation Transcription: Text Transcription: y’ y y’ x y 6 x^{4}+y^{5}=2x

(a) Find \(y^{\prime}\) by implicit differentiation. (b) Solve the equation explicitly for \(y\) and differentiate to get \(y^{\prime}\) in terms of \(x\). (c) Check that your solutions to parts (a) and (b) are consistent by substituting the expression for \(y\) into your solution for part (a). \(\sqrt{x}+\sqrt{y}=1\) ________________ Equation Transcription: Text Transcription: y’ y y’ x y sqrt{x} + sqrt{y}=1

(a) Find \(y^{\prime}\) by implicit differentiation. (b) Solve the equation explicitly for \(y\) and differentiate to get \(y^{\prime}\) in terms of \(x\). (c) Check that your solutions to parts (a) and (b) are consistent by substituting the expression for \(y\) into your solution for part (a). \(\frac{2}{x}-\frac{1}{y}=4\) ________________ Equation Transcription: Text Transcription: y’ y y’ x y frac{2}{x} - frac{1}{y} = 4

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

Find dy/dx by implicit differentiation. \(2 x^{2}+x y-y^{2}=2\)

Find \(d y / d x\) by implicit differentiation. \(x^{4}+x^{2} y^{2}+y^{3}=5\) ________________ Equation Transcription: Text Transcription: dy/dx x^{4}+x^{2} y^{2}+y^{3}=5

Find dy/dx by implicit differentiation. \(x^{3}-x y^{2}+y^{3}=1\)

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

Find \(d y / d x\) by implicit differentiation. \(x e^{y}=x-y\) ________________ Equation Transcription: Text Transcription: dy/dx xe^{y} = x-y

Find \(d y / d x\) by implicit differentiation. \(\sin x+\cos y=2 x-3 y\) Equation Transcription: Text Transcription: dy/dx sin x + cos y = 2x - 3y

Find \(d y / d x\) by implicit differentiation. \(e^{x} \sin y=x+y\) Equation Transcription: Text Transcription: dy/dx e^{x} sin y = x + y

Find \(d y / d x\) by implicit differentiation. \(\sin (x+y)=\cos x+\cos y\) Equation Transcription: Text Transcription: dy/dx sin (x + y) = cos x + cos y

Find \(d y / d x\) by implicit differentiation. \(\tan (x-y)=2 x y^{3}+1\) Equation Transcription: Text Transcription: dy/dx tan(x - y) = 2xy^{3} + 1

Find \(d y / d x\) by implicit differentiation. \(y \cos x=x^{2}+y^{2}\) Equation Transcription: Text Transcription: dy/dx y cos x = x^{2} + y^{2}

Find \(d y / d x\) by implicit differentiation. \(\sin (x y)=\cos (x+y)\) Equation Transcription: Text Transcription: dy/dx sin(xy) = cos(x + y)

Find \(d y / d x\) by implicit differentiation. \(2 x e^{y}+y e^{x}=3\) ________________ Equation Transcription: Text Transcription: dy/dx 2 x e^{y}+y e^{x}=3

Find \(d y / d x\) by implicit differentiation. \(\sin x \cos y=x^{2}-5 y\) Equation Transcription: Text Transcription: dy/dx sin x cos y = x^{2} - 5y

Find \(d y / d x\) by implicit differentiation. \(\sqrt{x+y}=x^{4}+y^{4}\) ________________ Equation Transcription: Text Transcription: dy/dx sqrt{x+y} = x^{4}+y^{4}

Find dy/dx by implicit differentiation. \(x y=\sqrt{x^{2}+y^{2}}\)

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

Find \(d y / d x\) by implicit differentiation. \(\cos \left(x^{2}+y^{2}\right)=x e^{y}\) ________________ Equation Transcription: Text Transcription: dy/dx cos(x^{2} + y^{2}) =xe^{y}

If \(f(x)+x^{2}[f(x)]^{3}=10\) and f(1) = 2, find \(f^{\prime}(1)\).

If \(g(x)+x \sin g(x)=x^{2}\), find \(g^{\prime}(0)\). ________________ Equation Transcription: Text Transcription: g(x) + x sin g(x) =x^2 g'(0)

Regard as the independent variable and \(x\) as the dependent variable and use implicit differentiation to find \(dx/dy\). \(x^{4} y^{2}-x^{3} y+2 x y^{3}=0\) ________________ Equation Transcription: Text Transcription: x dx/dy x^4 y^2-x^3y+2xy^3=0

Regard \(y\) as the independent variable and \(x\) as the dependent variable and use implicit differentiation to find \(d x / d y\). \(y \sec x=x \tan y\) Equation Transcription: Text Transcription: y x dy/dx y sec x = x tan y

Use implicit differentiation to find an equation of the tangent line to the curve at the given point. \(y e^{\sin x}=x \cos y\), \((0,0)\) Equation Transcription: Text Transcription: ye^{sin x} = x cos y (0, 0)

Use implicit differentiation to find an equation of the tangent line to the curve at the given point. \(\tan (x+y)+\sec (x-y)=2\), \((\pi / 8, \pi / 8)\) Equation Transcription: Text Transcription: tan(x + y) + sec(x-y) = 2 (pi/8, pi/8)

Use implicit differentiation to find an equation of the tangent line to the curve at the given point. (astroid)

Use implicit differentiation to find an equation of the tangent line to the curve at the given point. \(y^{2}(6-x)=x^{3}, \quad(2, \sqrt{2}) \quad\) (cissoid of Diocles) ________________ Equation Transcription: Text Transcription: y^2(6-x)=x^3, (2,square root 2)

Use implicit differentiation to find an equation of the tangent line to the curve at the given point. \(x^{2}-x y-y^{2}=1, \quad(2,1) \quad\) (hyperbola) Equation Transcription: Text Transcription: x^2-xy-y^2=1, (2,1)

Use implicit differentiation to find an equation of the tangent line to the curve at the given point. \(x^{2}+2 x y+4 y^{2}=12, \quad(2,1) \quad\) (ellipse) Equation Transcription: Text Transcription: x^2+2xy+4y^2=12, (2,1)

Use implicit differentiation to find an equation of the tangent line to the curve at the given point. \(x^{2}+y^{2}=\left(2 x^{2}+2 y^{2}-x\right)^{2}, \quad\left(0, \frac{1}{2}\right) \quad\) (cardioid)

Use implicit differentiation to find an equation of the tangent line to the curve at the given point. \(x^{2} y^{2}=(y+1)^{2}\left(4-y^{2}\right), \quad(2 \sqrt{3}, 1)\) (conchoid of Nicomedes) ________________ Equation Transcription: Text Transcription: x^2 y^2=(y+1)^2(4-y^2), (2 square root 3,1)

Use implicit differentiation to find an equation of the tangent line to the curve at the given point. \(2\left(x^{2}+y^{2}\right)^{2}=25\left(x^{2}-y^{2}\right), \quad(3,1) \quad\) (lemniscate) ________________ Equation Transcription: Text Transcription: 2(x^2+y^2)^2=25(x^2-y^2), (3,1)

Use implicit differentiation to find an equation of the tangent line to the curve at the given point. \(y^{2}\left(y^{2}-4\right)=x^{2}\left(x^{2}-5\right), \quad(0,-2) \quad\) (devil's curve) ________________ Equation Transcription: Text Transcription: y^2(y^2-4)=x^2(x^2-5), (0,-2)

(a) The curve with equation \(y^2=5 x^4-x^2\) is called a kampyle of Eudoxus. Find an equation of the tangent line to this curve at the point ( 1,2 ). (b) Illustrate part (a) by graphing the curve and the tangent line on a common screen. (Graph the implicitly defined curve if possible, or you can graph the upper and lower halves separately.)

(a) The curve with equation \(y^{2}=x^{3}+3 x^{2}\) is called the Tschirnhausen cubic. Find an equation of the tangent line to this curve at the point \((1,2)\). (b) At what points does this curve have horizontal tangents (c) Illustrate parts (a) and (b) by graphing the curve and the tangent lines on a common screen. Equation Transcription: Text Transcription: y^2=x^3+3x^2 (1,2)

Find \(y^{\prime \prime}\) by implicit differentiation. Simplify where possible. \(x^{2}+4 y^{2}=4\) Equation Transcription: Text Transcription: y'' x^2+4y^2=4

Find \(y^{\prime \prime}\) by implicit differentiation. Simplify where possible. \(x^{2}+x y+y^{2}=3\) ________________ Equation Transcription: Text Transcription: y'' x^2+xy+y^2=3

Find \(y^{\prime\prime}\) by implicit differentiation. Simplify where possible. \(\sin ?y+\cos ?x=1\) ________________ Equation Transcription: Text Transcription: y'' sin? y+cos ?x=1

Find \(y^{\prime \prime}\) by implicit differentiation. Simplify where possible. \(x^{3}-y^{3}=7\) ________________ Equation Transcription: Text Transcription: y'' x^3-y^3=7

If \(x y+e^{y}=e\), find the value of \(y^{\prime \prime}\) at the point where \(x=0\). Equation Transcription: Text Transcription: xy+e^y=e y'' x=0

If \(x^{2}+x y+y^{3}=1\), find the value of \(y^{\prime \prime \prime}\) at the point where \(x=1\). Equation Transcription: Text Transcription: x^2+xy+y^3=1 y''' x=1

Fanciful shapes can be created by using software that can graph implicitly defined curves. (a) Graph the curve with equation \(y\left(y^2-1\right)(y-2)=x(x-1)(x-2)\) At how many points does this curve have horizontal tangents? Estimate the x-coordinates of these points. (b) Find equations of the tangent lines at the points (0, 1) and (0, 2). (c) Find the exact x-coordinates of the points in part (a). (d) Create even more fanciful curves by modifying the equation in part (a).

(a) The curve with equation \(2 y^{3}+y^{2}-y^{5}=x^{4}-2 x^{3}+x^{2}\) has been likened to a bouncing wagon. Graph this curve and discover why. (b) At how many points does this curve have horizontal tangent lines? Find the \(x\)-coordinates of these points. Equation Transcription: Text Transcription: 2y^3+y^2-y^5=x^4-2x^3+x^2 x

Find the points on the lemniscate in Exercise 35 where the tangent is horizontal.

Show by implicit differentiation that the tangent line to the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) at the point \(\left(x_{0}, y_{0}\right)\) has equation \(\frac{x_{0} x}{a^{2}}+\frac{y_{0} y}{b^{2}}=1\) Equation Transcription: Text Transcription: x^2/a^2+y^2/b^2=1 (x_0,y_0) x_0 x/a^2+y_0 y/b^2=1

Find an equation of the tangent line to the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\) at the point \(\left(x_{0}, y_{0}\right)\). ________________ Equation Transcription: Text Transcription: x^2/a^2-y^2/b^2=1 (x_0,y_0)

Show that the sum of the x - and y-intercepts of any tangent line to the curve \(\sqrt{x}+\sqrt{y}=\sqrt{c}\) is equal to c.

Show, using implicit differentiation, that any tangent line at a point \(P\) to a circle with center \(O\) is perpendicular to the radius \(OP\). Equation Transcription: Text Transcription: P O OP

The Power Rule can be proved using implicit differentiation for the case where \(n\) is a rational number, \(n=p / q\), and \(y=f(x)=x^{n}\) is assumed beforehand to be a differentiable function. If \(y=x^{p / q}\), then \(y^{q}=x^{p}\). Use implicit differentiation to show that \(y^{\prime}=\frac{p}{q} x^{(p / q)-1}\) Equation Transcription: Text Transcription: n n=p/q y=f(x)=x^n y=xp/q y^q=x^p y'=p/q^x(p/q)-1

Orthogonal Trajectories Two curves are orthogonal if their tangent lines are perpendicular at each point of intersection. Show that the given families of curves are orthogonal trajectories of each other; that is, every curve in one family is orthogonal to every curve in the other family. Sketch both families of curves on the same axes. \(x^{2}+y^{2}=r^{2}, \quad a x+b y=0\) ________________ Equation Transcription: Text Transcription: x^2+y^2=r^2, ax+by=0

Orthogonal Trajectories Two curves are orthogonal if their tangent lines are perpendicular at each point of intersection. Show that the given families of curves are orthogonal trajectories of each other; that is, every curve in one family is orthogonal to every curve in the other family. Sketch both families of curves on the same axes. \(x^{2}+y^{2}=a x, \quad x^{2}+y^{2}=b y\) Equation Transcription: Text Transcription: x^2+y^2=ax, x^2+y^2=by

Orthogonal Trajectories Two curves are orthogonal if their tangent lines are perpendicular at each point of intersection. Show that the given families of curves are orthogonal trajectories of each other; that is, every curve in one family is orthogonal to every curve in the other family. Sketch both families of curves on the same axes. \(y=c x^{2}, \quad x^{2}+2 y^{2}=k\) Equation Transcription: Text Transcription: y=cx^2, x^2+2y^2=k

Orthogonal Trajectories Two curves are orthogonal if their tangent lines are perpendicular at each point of intersection. Show that the given families of curves are orthogonal trajectories of each other; that is, every curve in one family is orthogonal to every curve in the other family. Sketch both families of curves on the same axes. \(y=a x^{3}, \quad x^{2}+3 y^{2}=b\) Equation Transcription: Text Transcription: y=ax^3, ^2+3y^2=b

Show that the ellipse \(x^{2} / a^{2}+y^{2} / b^{2}=1\) and the hyperbola \(x^{2} / A^{2}-y^{2} / B^{2}=1\) are orthogonal trajectories if \(A^{2}<a^{2}\) and \(a^{2}-b^{2}=A^{2}+B^{2}\) (so the ellipse and hyperbola have the same foci). Equation Transcription: Text Transcription: x^2/a^2+y^2/b^2=1 x^2/A^2-y^2/B^2=1 A^2<a^2 a^2-b^2=A^2+B^2

Find the value of the number \(a\) such that the families of curves \(y=(x+c)^{-1}\) and \(y=a(x+k)^{1 / 3}\) are orthogonal trajectories. Equation Transcription: Text Transcription: y=(x+c)^-1 y=a(x+k)^1/3 a

The van der Waals equation for n moles of a gas is \(\left(P+\frac{n^2 a}{V^2}\right)(V-n b)=n R T\) where P is the pressure, V is the volume, and T is the temperature of the gas. The constant R is the universal gas constant and a and b are positive constants that are characteristic of a particular gas. (a) If T remains constant, use implicit differentiation to find dV/dP. (b) Find the rate of change of volume with respect to pressure of 1 mole of carbon dioxide at a volume of \(V=10 \mathrm{~L}\) and a pressure of \(P=2.5 \mathrm{~atm}\). Use \(a=3.592 \mathrm{~L}^2-\mathrm{atm} / \mathrm{mole}^2\) and \(b=0.04267 \mathrm{~L} / \mathrm{mole}\).

(a) Use implicit differentiation to find \(y^{\prime}\) if \(x^{2}+x y+y^{2}+1=0\) (b) Plot the curve in part (a). What do you see? Prove that what you see is correct. (c) In view of part (b), what can you say about the expression for \(y^{\prime}\) that you found in part (a)? Equation Transcription: Text Transcription: x^2+xy+y^2+1=0 y

The equation \(x^{2}-x y+y^{2}=3\) represents a "rotated ellipse," that is, an ellipse whose axes are not parallel to the coordinate axes. Find the points at which this ellipse crosses the x-axis and show that the tangent lines at these points are parallel.

(a) Where does the normal line to the ellipse \(x^{2}-x y+y^{2}=3\) at the point \((-1,1)\) intersect the ellipse a second time? (b) Illustrate part (a) by graphing the ellipse and the normal line. Equation Transcription: Text Transcription: x^2-xy+y^2=3 (-1,1)

Find all points on the curve \(x^{2} y^{2}+x y=2\) where the slope of the tangent line is \(-1\). Equation Transcription: Text Transcription: x^2 y^2+xy=2 -1

Find equations of both the tangent lines to the ellipse \(x^{2}+4 y^{2}=36\) that pass through the point \((12,3)\) Equation Transcription: Text Transcription: x^2+4y^2=36 (12,3)

Use implicit differentiation to find \(d y / d x\) for the equation \(\frac{x}{y}=y^{2}+1 \quad y \neq 0\) and for the equivalent equation \(x=y^{3}+y \quad y \neq 0\) Show that although the expressions you get for \(d y / d x\) look different, they agree for all points that satisfy the given equation. Equation Transcription: Text Transcription: dy/dx x/y=y^2+1 y not = 0 x=y^3+y y not = 0

The Bessel function of order \(0, y=J(x)\), satisfies the differential equation \(x y^{\prime \prime}+y^{\prime}+x y=0\) for all values of \(x\) and its value at 0 is \(J(0)=1\) (a) Find \(J^{\prime}(0)\) (b) Use implicit differentiation to find \(J^{\prime \prime}(0)\). Equation Transcription: 0 Text Transcription: 0,y=J(x) xy''+y'+xy=0 0 J(0)=1 J'(0) J''(0)

The figure shows a lamp located three units to the right of the \(y\)-axis and a shadow created by the elliptical region \(x^{2}+4 y^{2} \leqslant 5\). If the point \((-5,0)\) is on the edge of the shadow, how far above the \(x\)-axis is the lamp located? Equation Transcription: ? Text Transcription: y x^2+4y^2 less than or equal slant 5 (-5,0) x