Orthogonal Trajectories The figure below shows the topographic map carried by a group of

In Exercises 116, find by implicit differentiation. 10x21

In Exercises 116, find by implicit differentiation. x21y

In Exercises 116, find by implicit differentiation. 21y

In Exercises 116, find by implicit differentiation. 1y1

In Exercises 116, find by implicit differentiation. 1y1x

In Exercises 116, find by implicit differentiation. y1xx

In Exercises 116, find by implicit differentiation. 1xx

In Exercises 116, find by implicit differentiation. 1xxy

In Exercises 116, find by implicit differentiation. xxy2

In Exercises 116, find by implicit differentiation. xy2

In Exercises 116, find by implicit differentiation. y24

In Exercises 116, find by implicit differentiation. y24x

In Exercises 116, find by implicit differentiation. 24x

In Exercises 116, find by implicit differentiation. 4xy

In Exercises 116, find by implicit differentiation. 4xy3

In Exercises 116, find by implicit differentiation. xy3y

In Exercises 1720, (a) find two explicit functions by solving the equation for in terms of (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 and show that the result is equivalent to that of part (c). y3y

In Exercises 1720, (a) find two explicit functions by solving the equation for in terms of (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 and show that the result is equivalent to that of part (c). y3y

In Exercises 1720, (a) find two explicit functions by solving the equation for in terms of (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 and show that the result is equivalent to that of part (c). 3y9

In Exercises 1720, (a) find two explicit functions by solving the equation for in terms of (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 and show that the result is equivalent to that of part (c). y9

In Exercises 2128, find by implicit differentiation and evaluate the derivative at the given point. 9x

In Exercises 2128, find by implicit differentiation and evaluate the derivative at the given point. 9x2

In Exercises 2128, find by implicit differentiation and evaluate the derivative at the given point. 9x2y

In Exercises 2128, find by implicit differentiation and evaluate the derivative at the given point. x2y

In Exercises 2128, find by implicit differentiation and evaluate the derivative at the given point. x2yx

In Exercises 2128, find by implicit differentiation and evaluate the derivative at the given point. 2yx

In Exercises 2128, find by implicit differentiation and evaluate the derivative at the given point. yxx

In Exercises 2128, find by implicit differentiation and evaluate the derivative at the given point. xx

Famous Curves In Exercises 2932, find the slope of the tangent line to the graph at the given point. xx2

Famous Curves In Exercises 2932, find the slope of the tangent line to the graph at the given point. x2y

Famous Curves In Exercises 2932, find the slope of the tangent line to the graph at the given point. x2y

Famous Curves In Exercises 2932, find the slope of the tangent line to the graph at the given point. 2yx

Famous Curves In Exercises 3340, find an equation of the tangent line to the graph at the given point. To print an enlarged copy of the graph, select the MathGraph button. 2yx3

Famous Curves In Exercises 3340, find an equation of the tangent line to the graph at the given point. To print an enlarged copy of the graph, select the MathGraph button. yx3

Famous Curves In Exercises 3340, find an equation of the tangent line to the graph at the given point. To print an enlarged copy of the graph, select the MathGraph button. x34

Famous Curves In Exercises 3340, find an equation of the tangent line to the graph at the given point. To print an enlarged copy of the graph, select the MathGraph button. x34x

Famous Curves In Exercises 3340, find an equation of the tangent line to the graph at the given point. To print an enlarged copy of the graph, select the MathGraph button. 34xy

Famous Curves In Exercises 3340, find an equation of the tangent line to the graph at the given point. To print an enlarged copy of the graph, select the MathGraph button. 4xy

Famous Curves In Exercises 3340, find an equation of the tangent line to the graph at the given point. To print an enlarged copy of the graph, select the MathGraph button. 4xyx

Famous Curves In Exercises 3340, find an equation of the tangent line to the graph at the given point. To print an enlarged copy of the graph, select the MathGraph button. xyx3

(a) Use implicit differentiation to find an equation of the tangent line to the ellipse at (b) Show that the equation of the tangent line to the ellipse yx3

(a) Use implicit differentiation to find an equation of the tangent line to the hyperbola at (b) Show that the equation of the tangent line to the hyperbola x32

In Exercises 43 and 44, find implicitly and find the largest interval of the form or such that is a differentiable function of Write as a function of x32y

In Exercises 43 and 44, find implicitly and find the largest interval of the form or such that is a differentiable function of Write as a function of 32y

In Exercises 4550, find in terms of and 2y2

In Exercises 4550, find in terms of and 2y2x

In Exercises 4550, find in terms of and y2x2

In Exercises 4550, find in terms of and 2x2

In Exercises 4550, find in terms of and 2x2x

In Exercises 4550, find in terms of and x2xy

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 2xy

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 xy

In Exercises 53 and 54, find equations for the tangent line and normal line to the circle at the given points. (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. xyx

In Exercises 53 and 54, find equations for the tangent line and normal line to the circle at the given points. (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. yx

Show that the normal line at any point on the circle passes through the origin. x3

Two circles of radius 4 are tangent to the graph of at the point Find equations of these two circles. x3

In Exercises 57 and 58, find the points at which the graph of the equation has a vertical or horizontal tangent line. x32

In Exercises 57 and 58, find the points at which the graph of the equation has a vertical or horizontal tangent line. 32y

Orthogonal Trajectories In Exercises 5962, 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.] 32y

Orthogonal Trajectories In Exercises 5962, 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.] 2yx

Orthogonal Trajectories In Exercises 5962, 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.] 2yx2

Orthogonal Trajectories In Exercises 5962, 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.] yx2

Orthogonal Trajectories In Exercises 63 and 64, verify that the two families of curves are orthogonal where and are real numbers. Use a graphing utility to graph the two families for two values of and two values of x23

Orthogonal Trajectories In Exercises 63 and 64, verify that the two families of curves are orthogonal where and are real numbers. Use a graphing utility to graph the two families for two values of and two values of x23y

In Exercises 6568, differentiate (a) with respect to ( is a function of ) and (b) with respect to ( and are functions of ) 23y

In Exercises 6568, differentiate (a) with respect to ( is a function of ) and (b) with respect to ( and are functions of ) 3y1

In Exercises 6568, differentiate (a) with respect to ( is a function of ) and (b) with respect to ( and are functions of ) 3y1

In Exercises 6568, differentiate (a) with respect to ( is a function of ) and (b) with respect to ( and are functions of ) y1x

Describe the difference between the explicit form of a function and an implicit equation. Give an example of each. 1x2

Describe the difference between the explicit form of a function and an implicit equation. Give an example of each. 1x2y

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 and if they start from point 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, select the MathGraph button. x2y

Weather Map The weather map shows several isobars curves that represent areas of constant air pressure. Three high pressures and one low pressure 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. x2y2

Consider the equation (a) Use a graphing utility to graph the equation. (b) Find and graph the four tangent lines to the curve for (c) Find the exact coordinates of the point of intersection of the two tangent lines in the first quadrant. 2y23

Let be any tangent line to the curve Show that the sum of the and intercepts of is y23x

Prove (Theorem 2.3) that for the case in which is a rational number. (Hint: Write in the form and differentiate implicitly. Assume that and are integers, where 23x

Slope Find all points on the circle where the slope is 23x1

Horizontal Tangent Determine the point(s) at which the graph of has a horizontal tangent. 3x1y

Tangent Lines Find equations of both tangent lines to the ellipse that passes through the point x1y

Normals to a Parabola The graph shows the normal lines from the point to the graph of the parabola How many normal lines are there from the point to the graph of the parabola if (a) (b) and (c) For what value of are two of the normal lines perpendicular to each other? 1y1

Normal Lines (a) Find an equation of the normal line to the ellipse at the point (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? 1y12