Solved: In Exercises 53 and 54, find equations for the tangent line and normal line to

In Exercises 1-16, find dy/dx by implicit differentiation. \(x^2 + y^2 = 9\) Text Transcription: x^2 + y^2 = 9

In Exercises 1-16, find dy/dx by implicit differentiation. \(x^2-y^2=25\) Text Transcription: x^2 - y^2 = 25

In Exercises 1-16, find dy/dx by implicit differentiation. \(x^{1/2}+y^{1/2}=16\) Text Transcription: x^1/2 + y^1/2 =16

In Exercises 1-16, find dy/dx by implicit differentiation. \(x^3+y^3=64\) Text Transcription: x^3 + y^3 = 64

In Exercises 1-16, find dy/dx by implicit differentiation. \(x^3-xy+y^2=7\) Text Transcription: x^3 -xy + y^2 = 7

In Exercises 1-16, find dy/dx by implicit differentiation. \(x^2y+y^2x=-2\) Text Transcription: x^2 y + y^2 x = -2

In Exercises 1-16, find dy/dx by implicit differentiation. \(x^3y^3-y=x\) Text Transcription: x^3 y^3 - y = x

In Exercises 1-16, find dy/dx by implicit differentiation. \(\sqrt{xy}=x^2y+1\) Text Transcription: sqrtxy = x^2 y + 1

In Exercises 1-16, find dy/dx by implicit differentiation. \(x^3-3x^2y+2xy^2=12\) Text Transcription: x^3 - 3x^2 y + 2xy^2 = 12

In Exercises 1-16, find dy/dx by implicit differentiation. 4 cos x sin y = 1

In Exercises 1-16, find dy/dx by implicit differentiation. sin x + 2 cos 2y = 1

In Exercises 1-16, find dy/dx by implicit differentiation. \((\sin \pi x+\cos \pi y)^2=2\) Text Transcription: (sin pi x + cos pi y)^2 = 2

In Exercises 1-16, find dy/dx by implicit differentiation. sin x = x(1 + tan y)

In Exercises 1-16, find dy/dx by implicit differentiation. cot y = x - y

In Exercises 1-16, find dy/dx by implicit differentiation. y = sin xy

In Exercises 1-16, find dy/dx by implicit differentiation. \(x=\sec \frac{1}{y}\) Text Transcription: x=sec 1/y

In Exercises 17-20, (a) find two explicit functions by solving the equation for y in terms of x, (b) sketch the graph of the equation and label the parts given by the corresponding explicit functions, (c) differentiate the explicit functions, and (d) find dy/dx and show that the result is equivalent to that of part (c). \(x^2+y^2=64\) Text Transcription: x^2 + y^2 = 64

In Exercises 17-20, (a) find two explicit functions by solving the equation for y in terms of x, (b) sketch the graph of the equation and label the parts given by the corresponding explicit functions, (c) differentiate the explicit functions, and (d) find dy/dx and show that the result is equivalent to that of part (c). \(x^2+y^2-4x+6y+9=0\) Text Transcription: x^2 + y^2 - 4x + 6y + 9 = 0

In Exercises 17-20, (a) find two explicit functions by solving the equation for y in terms of x, (b) sketch the graph of the equation and label the parts given by the corresponding explicit functions, (c) differentiate the explicit functions, and (d) find dy/dx and show that the result is equivalent to that of part (c). \(16x^2+25y^2=400\) Text Transcription: 16x^2 + 25y^2 = 400

In Exercises 17-20, (a) find two explicit functions by solving the equation for y in terms of x, (b) sketch the graph of the equation and label the parts given by the corresponding explicit functions, (c) differentiate the explicit functions, and (d) find dy/dx and show that the result is equivalent to that of part (c). \(16y^2-x^2=16\) Text Transcription: 16y^2 - x^2 = 16

In Exercises 21-28, find dy/dx by implicit differentiation and evaluate the derivative at the given point. xy = 6, (-6, -1)

In Exercises 21-28, find dy/dx by implicit differentiation and evaluate the derivative at the given point. \(x^2-y^3=0,\ \ \ (1,\ 1)\) Text Transcription: x^2 - y^3 = 0, (1, 1)

In Exercises 21-28, find dy/dx by implicit differentiation and evaluate the derivative at the given point. \(y^2=\frac{x^2 -49}{x^2 +49},\ \ \ (7,\ 0)\) Text Transcription: y^2 = x^2 -49/x^2 +49, (7, 0)

In Exercises 21-28, find dy/dx by implicit differentiation and evaluate the derivative at the given point. \((x+y)^3=x^3+y^3,\ \ \ (-1,\ 1)\) Text Transcription: (x+y)^3 = x^3 + y^3 , (-1, 1)

In Exercises 21-28, find dy/dx by implicit differentiation and evaluate the derivative at the given point. \(x^{2/3}+y^{2/3}=5,\ \ \ (8,\ 1)\) Text Transcription: x^2/3 + y^2/3 = 5, (8, 1)

In Exercises 21-28, find dy/dx by implicit differentiation and evaluate the derivative at the given point. \(x^3 + y^3=6xy-1,\ \ \ (2,\ 3)\) Text Transcription: x^3 + y^3 = 6xy-1, (2, 3)

In Exercises 21-28, find dy/dx by implicit differentiation and evaluate the derivative at the given point. tan(x + y) = x, (0, 0)

In Exercises 21-28, find dy/dx by implicit differentiation and evaluate the derivative at the given point. \(x \cos y=1,\ \ \ (2,\ \frac{\pi}{3})\) Text Transcription: x cos y = 1, (2, pi/3)

Famous Curves In Exercises 29-32, find the slope of the tangent line to the graph at the given point. Witch of Agnesi: \((x^2 +4)y = 8\) Point: (2, 1) Text Transcription: (x^2 + 4)y = 8

Famous Curves In Exercises 29-32, find the slope of the tangent line to the graph at the given point. Cissoid: \((4-x)y^2=x^3\) Point: (2, 2) Text Transcription: (4-x)y^2 = x^3

Famous Curves In Exercises 29-32, find the slope of the tangent line to the graph at the given point. Bifolium: \((x^2+y^2)^2=4x^2y\) Point: (1, 1) Text Transcription: (x^2 + y^2)^2 = 4x^2 y

Famous Curves In Exercises 29-32, find the slope of the tangent line to the graph at the given point. Folium of Descartes: \(x^3+y^3-6xy=0\) Point: \((\frac{4}{3},\ \frac{8}{3})\) Text Transcription: x^3 + y^3 - 6xy = 0 (4/3 , 8/3)

Famous Curves In Exercises 33-40, find an equation of the tangent line to the graph at the given point. To print an enlarged copy of the graph, go to the website www.mathgraphs.com. Parabola

Famous Curves In Exercises 33-40, find an equation of the tangent line to the graph at the given point. To print an enlarged copy of the graph, go to the website www.mathgraphs.com. Circle

Famous Curves In Exercises 33-40, find an equation of the tangent line to the graph at the given point. To print an enlarged copy of the graph, go to the website www.mathgraphs.com. Rotated hyperbola

Famous Curves In Exercises 33-40, find an equation of the tangent line to the graph at the given point. To print an enlarged copy of the graph, go to the website www.mathgraphs.com. Rotated ellipse

Famous Curves In Exercises 33-40, find an equation of the tangent line to the graph at the given point. To print an enlarged copy of the graph, go to the website www.mathgraphs.com. Cruciform

Famous Curves In Exercises 33-40, find an equation of the tangent line to the graph at the given point. To print an enlarged copy of the graph, go to the website www.mathgraphs.com. Astroid

Famous Curves In Exercises 33-40, find an equation of the tangent line to the graph at the given point. To print an enlarged copy of the graph, go to the website www.mathgraphs.com. Lemniscate

Famous Curves In Exercises 33-40, find an equation of the tangent line to the graph at the given point. To print an enlarged copy of the graph, go to the website www.mathgraphs.com. Kappa curve

(a) Use implicit differentiation to find an equation of the tangent line to the ellipse \(\frac{x^2}{2}+\frac{y^2}{8}=1\) at (1, 2). (b) Show that the equation of the tangent line to the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) at \((x_0,\ y_0)\) is \(\frac{x_0x}{a^2}+\frac{y_0y}{b^2}=1\). Text Transcription: x^2 /2 + y^2 /8 = 1 x^2 /a^2 +y^2 /b^2 =1 (x_0 , y_0) x_0 x/a^2 + y_0 y/b^2 =1

(a) Use implicit differentiation to find an equation of the tangent line to the hyperbola \(\frac{x^2}{6}-\frac{y^2}{8}=1\) at (3, -2). (b) Show that the equation of the tangent line to the hyperbola \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\) at \((x_0,\ y_0)\) is \(\frac{x_0x}{a^2}-\frac{y_0y}{b^2}=1\). Text Transcription: x^2 /6 - y^2 /8 =1 x^2 /a^2 - y^2 /b^2 (x_0 , y_0) x_0 x/a^2 - y_0 y/b^2 =1

In Exercises 43 and 44, find dy/dx implicitly and find the largest interval of the form -a < y < a or 0 < y < a such that y is a differentiable function of x. Write dy/dx as a function of x. tan y = x

In Exercises 43 and 44, find dy/dx implicitly and find the largest interval of the form -a < y < a or 0 < y < a such that y is a differentiable function of x. Write dy/dx as a function of x. cos y = x

In Exercises 45-50, find \(d^2y/dx^2\) in terms of x and y. \(x^2+y^2=4\) Text Transcription: d^2 y/dx^2 x^2 +y^2 =4

In Exercises 45-50, find \(d^2y/dx^2\) in terms of x and y. \(x^2y^2-2x=3\) Text Transcription: d^2 y/dx^2 x^2 y^2 -2x =3

In Exercises 45-50, find \(d^2y/dx^2\) in terms of x and y. \(x^2-y^2=36\) Text Transcription: d^2 y/dx^2 x^2 -y^2 =36

In Exercises 45-50, find \(d^2y/dx^2\) in terms of x and y. 1 - xy = x - y Text Transcription: d^2 y/dx^2

In Exercises 45-50, find \(d^2y/dx^2\) in terms of x and y. \(y^2=x^3\) Text Transcription: d^2 y/dx^2 y^2 =x^3

In Exercises 45-50, find \(d^2y/dx^2\) in terms of x and y. \(y^2=10x\) Text Transcription: d^2 y/dx^2 y^2 =10x

In Exercises 51 and 52, use a graphing utility to graph the equation. Find an equation of the tangent line to the graph at the given point and graph the tangent line in the same viewing window. \(\sqrt{x}+\sqrt{y}=5,\ \ \ (9,\ 4)\) Text Transcription: sqrtx + sqrty =5, (9, 4)

In Exercises 51 and 52, use a graphing utility to graph the equation. Find an equation of the tangent line to the graph at the given point and graph the tangent line in the same viewing window. \(y^2=\frac{x-1}{x^2+1},\ \ \ (2,\ \frac{\sqrt{5}}{5})\) Text Transcription: y^2 = x-1/x^2 +1 , (2, sqrt5/ 5)

In Exercises 53 and 54, find equations for the tangent line and normal line to the circle at each given point. (The normal line at a point is perpendicular to the tangent line at the point.) Use a graphing utility to graph the equation, tangent line, and normal line. \(x^2+y^2=25\) \((4,\ 3),\ (-3,\ 4)\) Text Transcription: x^2 +y^2 =25 (4, 3), (-3, 4)

In Exercises 53 and 54, find equations for the tangent line and normal line to the circle at each given point. (The normal line at a point is perpendicular to the tangent line at the point.) Use a graphing utility to graph the equation, tangent line, and normal line. \(x^2+y^2=36\) \((6,\ 0),\ (5,\ \sqrt{11})\) Text Transcription: x^2 +y^2 =36 (6, 0), (5, sqrt11)

Show that the normal line at any point on the circle \(x^2+y^2=r^2\) passes through the origin. Text Transcription: x^2 +y^2 =r^2

Two circles of radius 4 are tangent to the graph of \(y^2=4x\) at the point (1, 2). Find equations of these two circles. Text Transcription: y^2 =4x

In Exercises 57 and 58, find the points at which the graph of the equation has a vertical or horizontal tangent line. \(25x^2+16y^2+200x-160y+400=0\) Text Transcription: 25x^2 +16y^2 +200x -160y +400 =0

In Exercises 57 and 58, find the points at which the graph of the equation has a vertical or horizontal tangent line. \(4x^2+y^2-8x+4y+4=0\) Text Transcription: 4x^2 +y^2 -8x+4y=0

Orthogonal Trajectories In Exercises 59-62, use a graphing utility to sketch the intersecting graphs of the equations and show that they are orthogonal. [ Two graphs are orthogonal if at their point(s) of intersection their tangent lines are perpendicular to each other.] \(2x^2+y^2=6\) \(y^2=4x\) Text Transcription: 2x^2 +y^2 =6 y^2 =4x

Orthogonal Trajectories In Exercises 59-62, use a graphing utility to sketch the intersecting graphs of the equations and show that they are orthogonal. [ Two graphs are orthogonal if at their point(s) of intersection their tangent lines are perpendicular to each other.] \(y^2=x^3\) \(2x^2+3y^2=5\) Text Transcription: y^2 =x^3 2x^2 +3y^2 =5

Orthogonal Trajectories In Exercises 59-62, use a graphing utility to sketch the intersecting graphs of the equations and show that they are orthogonal. [ Two graphs are orthogonal if at their point(s) of intersection their tangent lines are perpendicular to each other.] x + y = 0 x = sin y

Orthogonal Trajectories In Exercises 59-62, use a graphing utility to sketch the intersecting graphs of the equations and show that they are orthogonal. [ Two graphs are orthogonal if at their point(s) of intersection their tangent lines are perpendicular to each other.] \(x^3=3(y-1)\) \(x(3y-29)=3\) Text Transcription: x^3 =3(y-1) x(3y-29)=3

Orthogonal Trajectories In Exercises 63 and 64, verify that the two families of curves are orthogonal where C and K are real numbers. Use a graphing utility to graph the two families for two values of C and two values of K. \(xy=C,\ \ \ x^2-y^2=K\) Text Transcription: xy=C, x^2 -y^2 =K

Orthogonal Trajectories In Exercises 63 and 64, verify that the two families of curves are orthogonal where C and K are real numbers. Use a graphing utility to graph the two families for two values of C and two values of K. \(x^2+y^2=C^2,\ \ \ y=Kx\) Text Transcription: x^2 +y^2 =C^2 , y=Kx

In Exercises 65-68, differentiate (a) with respect to x (y is a function of x) and (b) with respect to t (x and y are functions of t). \(2y^2-3x^4=0\) Text Transcription: 2y^2 -3x^4 =0

In Exercises 65-68, differentiate (a) with respect to x (y is a function of x) and (b) with respect to t (x and y are functions of t). \(x^2-3xy^2+y^3=10\) Text Transcription: x^2 -3xy^2 +y^3 =10

In Exercises 65-68, differentiate (a) with respect to x (y is a function of x) and (b) with respect to t (x and y are functions of t). \(\cos \pi y-3 \sin \pi x=1\) Text Transcription: cos pi y - 3 sin pi x = 1

In Exercises 65-68, differentiate (a) with respect to x (y is a function of x) and (b) with respect to t (x and y are functions of t). 4 sin x cos y = 1

Describe the difference between the explicit form of a function and an implicit equation. Give an example of each.

In your own words, state the guidelines for implicit differentiation.

Orthogonal Trajectories The figure below shows the topographic map carried by a group of hikers. The hikers are in a wooded area on top of the hill shown on the map and they decide to follow a path of steepest descent (orthogonal trajectories to the contours on the map). Draw their routes if they start from point A and if they start from point B. If their goal is to reach the road along the top of the map, which starting point should they use? To print an enlarged copy of the graph, go to the website www.mathgraphs.com.

Weather MapThe weather map shows several isobars— curves that represent areas of constant air pressure. Three high pressures H and one low pressure L are shown on the map. Given that wind speed is greatest along the orthogonal trajectories of the isobars, use the map to determine the areas having high wind speed.

Consider the equation \(x^4=4(4x^2-y^2)\). (a) Use a graphing utility to graph the equation. (b) Find and graph the four tangent lines to the curve for y = 3. (c) Find the exact coordinates of the point of intersection of the two tangent lines in the first quadrant. Text Transcription: x^4 =4(4x^2 -y^2)

Determine if the statement is true. If it is false, explain why and correct it. For each statement, assume y is a function of x. (a) \(\frac{d}{dx}\cos(x^2)=-2x\sin (x^2)\) (b) \(\frac{d}{dy}\cos(y^2)=2y \sin(y^2)\) (c) \(\frac{d}{dx}\cos(y^2)=-2y\sin(y^2)\) Text Transcription: d/dx cos (x^2) = -2x sin (x^2) d/dy cos (y^2) = 2y sin (y^2) d/dx cos (y^2) = -2y sin (y^2)

Let L be any tangent line to the curve \(\sqrt{x}+\sqrt{y}=\sqrt{c}\). Show that the sum of the x- and y-intercepts of L is c. Text Transcription: sqrtx + sqrty = sqrtc

Prove (Theorem 2.3) that \(d/dx\ [x^n]=nx^{n-1}\) for the case in which n is a rational number. (Hint: Write \(y=x^{p/q}\) in the form \(y^q=x^p\) and differentiate implicitly. Assume that p and q are integers, where q > 0.) Text Transcription: d/dx [X^n] = nx^n-1 y=x^p/q y^q = x^p

Slope Find all points on the circle \(x^2+y^2=100\) where the slope is \(\frac{3}{4}\). Text Transcription: x^2 +y^2 =100 3/4

Horizontal Tangent Determine the point(s) at which the graph of \(y^4=y^2-x^2\) has a horizontal tangent. Text Transcription: y^4 = y^2 -x^2

Tangent Lines Find equations of both tangent lines to the ellipse \(\frac{x^2}{4}+\frac{y^2}{9}=1\) that passes through the point (4, 0). Text Transcription: x^2 /4 + y^2 /9 =1

Normals to a Parabola The graph shows the normal lines from the point (2, 0) to the graph of the parabola \(x=y^2\). How many normal lines are there from the point \((x_0,\ 0)\) to the graph of the parabola if (a) \(x_0=\frac{1}{4}\), (b) \(x_0=\frac{1}{2}\), and (c) \(x_0=1\)? For what value of \(x_0\) are two of the normal lines perpendicular to each other? Text Transcription: x=y^2 (x_0 , 0) x_0 = 1/4 x_0 = 1/2 x_0 = 1 x_0

Normal Lines (a) Find an equation of the normal line to the ellipse \(\frac{x^2}{32}+\frac{y^2}{8}=1\) at the point (4, 2). (b) Use a graphing utility to graph the ellipse and the normal line. (c) At what other point does the normal line intersect the ellipse? Text Transcription: x^2 /32 + y^2 /8 =1