(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 implicitly and show that the result is equivalent to that of part (c). x2 + y2 – 4x + 6y + 9 = 0
> 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 the path of steepest descent (orthogonal trajectories to the contours on the map). Dra
> 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. x2 + y2 = C2 y = Kx
> xy = C, x2 – y2 = k 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.
> x3 = 3(y - 1) x(3y - 29) = 3 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 othe
> 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
> 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.] y2 = x3 2x2 + 3y2 = 5
> Find the derivative of the function.
> 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.] 2x2 + y2 = 6 y2 = 4x
> 4x2 + y2 – 8x + 4y + 4 = 0 Find the points at which the graph of the equation has a vertical or horizontal tangent line.
> 25x2 + 16y2 + 200x – 160y + 400 = 0 Find the points at which the graph of the equation has a vertical or horizontal tangent line.
> Two circles of radius 4 are tangent to the graph of y2 = 4x at the point (1, 2). Find equations of these two circles.
> Show that the normal line at any point on the circle x2 + y2 = r2 passes through the origin.
> 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 circle, the tangent lines, and the normal lines. x2
> x2 + y2 = 25 (4, 3), (-3, 4) 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 circle, the tangent l
> 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.
> 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.
> 3xy − 4 cos x = −6 Find d2y/dx2 implicitly in terms of x and y.
> Find the derivative of the function.
> 7xy + sin x - 2 Find d2y/dx2 implicitly in terms of x and y.
> xy – 1 = 2x + y2 Find d2y/dx2 implicitly in terms of x and y.
> x2y – 2 = 5x + y Find d2y/dx2 implicitly in terms of x and y.
> x2y – 4x = 5 Find d2y/dx2 implicitly in terms of x and y.
> x2 + y2 = 4 Find d2y/dx2 implicitly in terms of x and y.
> 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
> 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
> Use implicit differentiation to find an equation of the Show that the equation of the tangent line to the ellipse
> Use implicit differentiation to find an equation of the Show that the equation of the tangent line to the ellipse
> Explain why the derivative of x2 + y2 + 2 = 1 does not mean anything.
> Find the derivative of the function.
> Write two different equations in implicit form that you can write in explicit form. Then write two different equations in implicit form that you cannot write in explicit form.
> Find an equation of the tangent line to the graph at the given point. Kappa curve
> Find an equation of the tangent line to the graph at the given point. Lemniscate
> Find an equation of the tangent line to the graph at the given point. Astroid
> Find an equation of the tangent line to the graph at the given point. Cruciform
> Find an equation of the tangent line to the graph at the given point. Circle
> Find an equation of the tangent line to the graph at the given point. Parabola
> Folium of Descartes: x3 + y3 - 6xy = 0 Find the slope of the tangent line to the graph at the given point.
> Find the slope of the tangent line to the graph at the given point. Bifolium: (x2 + y2)2 = 4x2y
> Find the slope of the tangent line to the graph at the given point. Cissoid: (4 - x)y2 = x3
> Find the derivative of the function.
> Find the slope of the tangent line to the graph at the given point. Witch of Agnesi: (x2 + 4)y = 8
> x cos y = 1, (2, π/3) Find dy/dx by implicit differentiation. Then find the slope of the graph at the given point.
> tan(x + y) = x, (0, 0) Find dy/dx by implicit differentiation. Then find the slope of the graph at the given point.
> x3 + y3 – 6xy – 1, (2, 3) Find dy/dx by implicit differentiation. Then find the slope of the graph at the given point.
> (x + y)3 = x3 + y3, (-1, 1) Find dy/dx by implicit differentiation. Then find the slope of the graph at the given point.
> Find dy/dx by implicit differentiation. Then find the slope of the graph at the given point.
> Find dy/dx by implicit differentiation. Then find the slope of the graph at the given point.
> 3x3y = 6, (1, 2) Find dy/dx by implicit differentiation. Then find the slope of the graph at the given point.
> xy = 6, (-6, -1) Find dy/dx by implicit differentiation. Then find the slope of the graph at the given point.
> Find the derivative of the function.
> (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 implicitly
> (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 implicitly
> (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 implicitly
> Find dy/dx by implicit differentiation.
> y = sin xy Find dy/dx by implicit differentiation.
> cot y = x - y Find dy/dx by implicit differentiation.
> csc x = x(1 + tan y) Find dy/dx by implicit differentiation.
> (sin πx + cos πy)2 = 2 Find dy/dx by implicit differentiation.
> sin x + 2 cos 2y = 1 Find dy/dx by implicit differentiation.
> x4y – 8xy + 3xy2 = 9 Find dy/dx by implicit differentiation.
> Find the derivative of the function.
> x3 – 3x2y + 2xy2 = 12 Find dy/dx by implicit differentiation.
> Find dy/dx by implicit differentiation.
> x3y3 – y - x = 0 Find dy/dx by implicit differentiation.
> x2y + y2x = -2 Find dy/dx by implicit differentiation.
> x3 – xy + y2 = 7 Find dy/dx by implicit differentiation.
> 2x2 + 3y3 = 64 Find dy/dx by implicit differentiation.
> x5 + y5 = 16 Find dy/dx by implicit differentiation.
> x2 - y2 = 25 Find dy/dx by implicit differentiation.
> x2 + y2 = 9 Find dy/dx by implicit differentiation.
> How is the Chain Rule applied when finding dy/dx implicitly?
> Find the derivative of the function.
> Explain when you have to use implicit differentiation to find a derivative.
> In your own words, state the guidelines for implicit differentiation.
> Describe the difference between the explicit form of a function and an implicit equation. Give an example of each.
> Let k be a fixed positive integer. The nth derivative polynomial. Find Pn (1)
> Let f(x) = q1 sin x + a2 sin 2x + ∙ ∙ ∙ + an sin nx, where a1, a2, . . ., an are real numbers and where n is a positive integer.
> Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If y is a differentiable function of u, u is a differentiable function of v, and v is a differentiable function of x, then.
> Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If y is a differentiable function of u, and u is a differentiable function of x, then y is a differentiable function of x..
> Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.
> The linear and quadratic approximations of a function f at x = a are P1 (x) = f’(a)(x - a) + f(a) and P2 (x) = ½ f’’(a)(x - a)2 + f’(a)(x - a) + f(a). (a)Find the spec
> f(t) = (9t + 2)2/3 Find the derivative of the function.
> The linear and quadratic approximations of a function f at x = a are P1 (x) = f’(a)(x - a) + f(a) and P2 (x) = ½ f’’(a)(x - a)2 + f’(a)(x - a) + f(a). (a)Find the spec
> f(x) = |sin x| Use the result of Exercise 114 to find the derivative of the function. Answer: f(x) = |sin x|
> h(x) = |x| cos x Use the result of Exercise 114 to find the derivative of the function. Answer: h(x) = |x| cos x
> f(x) = |x2 - 9| Use the result of Exercise 114 to find the derivative of the function.
> g(x) = |3x - 5| Use the result of Exercise 114 to find the derivative of the function.
> Let u be a differentiable function of x. Use the fact that
> Show that the derivative of an odd function is even. That is, if (-x) = -f(x), then f’(-x) = f’(x). Show that the derivative of an even function is odd. That is, if (-x) = -f(x), then f’(-x) = f’(x).
> Find the derivative of the function g(x) = sin2 x + cos2 x in two ways. For f(x) = sec2 x and g(x) = tan2 x, show that f’(x) = g’(x).
> Let r(x) = f(g(x)) and s(x) = g(f(x)), where f and g are shown in the figure. Find (a) r’(1) and (b) s’(4).
> Let f be a differentiable function of period p. Is the function f′ periodic? Verify your answer. Consider the function g(x) = f(2x). Is the function g′(x) periodic? Verify your answer.
> Find the derivative of the function. g(x) = 3(4 – 9x)5/6
> Consider the function f(x) = sin βx, where β is a constant. Find the first-, second-, third-, and fourth-order derivatives of the function. Verify that the function and its second derivative satisfy the equation f″(x) + β2 f(x) = 0. Use the results of pa
> The value V of a machine t years after it is purchased is inversely proportional to the square root of t + 1. The initial value of the machine is $10,000. Write V as a function of t. Find the rate of depreciation when t = 1. Find the rate of depreciation
> The number N of bacteria in a culture after t days is modeled by Find the rate of change of N with respect to t when t = 0, (b) t = 1, (c) t = 2, (d) t = 3, and (e) t = 4. (f) what can you conclude?
> The cost C (in dollars) of producing x units of a product is C = 60x + 1350. For one week, management determined that the number of units produced x at the end of t hours can be modeled by x = -1.6t3 + 19t2 – 0.5 t – 1
> The normal daily maximum temperatures T (in degrees Fahrenheit) for Chicago, Illinois, are shown in the table. Use a graphing utility to plot the data and find a model for the data of the form T(t) = a + b sin(ct - d) where T is the temperature and t is
> A buoy oscillates in simple harmonic motion y = A cos ωt as waves move past it. The buoy moves a total of 3.5 feet (vertically) from its low point to its high point. It returns to its high point every 10 seconds. Write an equation describing the motion o
> A 15-centimeter pendulum moves according to the equation θ = 0.2 cos 8t, where θ is the angular displacement from the vertical in radians and t is the time in seconds. Determine the maximum angular displacement and the rate of change of θ when t = 3 seco
