Find R'(0), R(x) = x + 3x3 + 5x5/1 + 3x3 + 6x6 + 9x9 where Hint: Instead of finding R’(x) first, let f (x) be the numerator and g (x) the denominator of R (x) and compute R' (0) from f (0), f' (0), g (0), and g' (0).
> 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 oth
> 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 oth
> 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 oth
> 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 oth
> Find an equation of the tangent line to the curve at the given point. y = (1 + 2x)10, (0, 1)
> Show by implicit differentiation that the tangent to the ellipse x2/a2 + y2/b2 = 1 at the point (x0, y0) is x0x/a2 + y0y/b2 = 1
> Find dy/dx by implicit differentiation. 2√x + √y = 3
> Find y' and y". y = eax sin βx
> Find y' and y". y = cos2x
> Find y' and y". y = cos (x2)
> Find the derivative of the function. y = 23x2
> Zoom in toward the points (1, 0), (0, 1), and (-1, 0) on the graph of the function g (x) = (x2 – 1)2/3. What do you notice? Account for what you see in terms of the differentiability of g.
> A particle moves according to a law of motion s = f (t), t > 0, where is measured in seconds and in feet. (a). Find the velocity at time t. (b). What is the velocity after 3 s? (c). When is the particle at rest? (d). When is the particle moving in the po
> Find the derivative of the function. y = cos √sin (tan πx)
> Find y" by implicit differentiation. x4 + y4 = a4
> Find y" by implicit differentiation. x3 + y3 = 1
> Find y" by implicit differentiation. √x + √y = 1
> Find y" by implicit differentiation. 9x2 + y2 = 9
> Find the derivative of the function. f (t) = √t/t2 + 4
> Find dy/dx by implicit differentiation. x3 + y3 = 1
> (a). The curve with equation y2 = 5x4 – x2 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. (If your graphing d
> Use implicit differentiation to find an equation of the tangent line to the curve at the given point. у? - 4) — х"({? - 5) К0, —2) (devil's curve) y
> Use implicit differentiation to find an equation of the tangent line to the curve at the given point. 2(x? + y?)? = 25(x² – y²) (3, 1) (lemniscate) y4
> Find the points on the lemniscate in Exercise 27 where the tangent is horizontal. Exercise 27: Use implicit differentiation to find an equation of the tangent line to the curve at the given point. 2(x? + y?)? = 25(x² – y²) (3, 1) (lemniscate) y4
> Use implicit differentiation to find an equation of the tangent line to the curve at the given point. x2/3 + y2/3 = 4 (-3 √3, 1) (astroid) 8
> Use implicit differentiation to find an equation of the tangent line to the curve at the given point. x2 + y2 = (2x2 + 2y2) – x)2 (0, 1 /2) (cardioid) y.
> Use implicit differentiation to find an equation of the tangent line to the curve at the given point. x² + 2xy – y² + x = 2, (1, 2) (hyperbola)
> Use implicit differentiation to find an equation of the tangent line to the curve at the given point. x² + xy + y? = 3, (1, 1) (ellipse)
> Use implicit differentiation to find an equation of the tangent line to the curve at the given point. sin(x + y) = 2x – 2y, (7, 7) %3D
> Use implicit differentiation to find an equation of the tangent line to the curve at the given point. y sin 2x = x cos 2y, (7/2, 7/4)
> Regard as the independent variable and as the dependent variable and use implicit differentiation to find dx/dy. y sec x = x tan y
> Write the composite function in the form f (g (x)). [Identify the inner function u = g (x) and the outer function y = f (u).] Then find the derivative dy/dx. y = (2x3 + 5)4
> Regard as the independent variable and as the dependent variable and use implicit differentiation to find dx/dy. x4y2 - x3y + 2xy3 = 0
> If g (x) + x sin g (x) = x2, find g'(0).
> (a). The curve with equation has been likened to a bouncing wagon. Use a computer algebra system to 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.
> If f (x) + x2 [f (x)]3 = 10 and f (1) = 2, find f'(1).
> Find dy/dx by implicit differentiation. sin x + cos y = sin x cos y
> Find dy/dx by implicit differentiation. ey cos x = 1 + sin (xy)
> Find dy/dx by implicit differentiation. tan (x – y) = y/1 + xy2
> Find dy/dx by implicit differentiation. ex/y = x - y
> Find dy/dx by implicit differentiation. y sin (x2) = x sin (y2)
> Find dy/dx by implicit differentiation. 4 cos x sin y = 4
> Find dy/dx by implicit differentiation. 1 + x = sin (xy2)
> (a). Find y' by implicit differentiation. (b). Solve the equation explicitly for and differentiate to get y' in terms of x. (c). Check that your solutions to parts (a) and (b) are consistent by substituting the expression for into your solution for part
> Find the derivative of the function. F (x) = √1 - 2x
> Fanciful shapes can be created by using the implicit plotting capabilities of computer algebra systems. (a). Graph the curve with equation At, how many points does this curve have horizontal tangents? Estimate the -coordinates of these points. (b). Fin
> Find the derivative of the function. F (x) = (4x - x2)100
> Find the derivative of the function. F (x) = (x4 + 3x2 – 2)5
> Write the composite function in the form f (g (x)). [Identify the inner function u = g (x) and the outer function y = f (u).] Then find the derivative dy/dx. y = √2 - ex
> Let f and be the functions in Exercise 53. Exercise 53: A table of values for f, g, f', and g' is given. (a). If F (x) = f (f (x)), find F'(2). (b). If G (x) = g (g (x)), find G'(3).
> A table of values for f, g, f', and g' is given. (a). If h (x) = f (g (x)), find h'(1). (b). If H (x) = g (f (x)), find H'(1).
> Find the x-coordinates of all points on the curve y = sin 2x – 2 sin x at which the tangent line is horizontal.
> Write the composite function in the form f (g (x)). [Identify the inner function u = g (x) and the outer function y = f (u).] Then find the derivative dy/dx. y = e√x
> (a). If f (x) = x√2 – x2, find f'(x). (b). Check to see that your answer to part (a) is reasonable by comparing the graphs of f and f'.
> (a). The curve y = |x|/√2 – x2 is called a bullet-nose curve. Find an equation of the tangent line to this curve at the point (1, 1). (b). Illustrate part (a) by graphing the curve and the tangent line on the same screen.
> Suppose that we don’t have a formula for g (x) but we know that g (2) = -4 and g'(x) = √x2 + 5 for all x. (a). Use a linear approximation to estimate g (1.95) and g (2.05). (b). Are your estimates in part (a) too large or too small? Explain.
> (a). Find an equation of the tangent line to the curve y = 2/ (1 + e-x) at the point (0, 1). (b). Illustrate part (a) by graphing the curve and the tangent line on the same screen.
> Find an equation of the tangent line to the curve at the given point. y = sin x + sin2x, (0, 0)
> Find an equation of the tangent line to the curve at the given point. y = sin (sin x), (π, 0)
> Find an equation of the tangent line to the curve at the given point. y = √1 + x3, (2, 3)
> Find y' and y". y = eex
> Write the composite function in the form f (g (x)). [Identify the inner function u = g (x) and the outer function y = f (u).] Then find the derivative dy/dx. y = sin (cot x)
> Find the derivative of the function. y = √(x+√(x+√x))
> Find the derivative of the function. y = cot2(sin θ)
> Find the derivative of the function. y = sin (sin (sin x))
> Find the derivative of the function. y = 2 simπx
> Suppose that the only information we have about a function f is that f (1) = 5 and the graph of its derivative is as shown. (a). Use a linear approximation to estimate f (0.9) and f (1.1). (b). Are your estimates in part (a) too large or too small? Exp
> Write the composite function in the form f (g (x)). [Identify the inner function u = g (x) and the outer function y = f (u).] Then find the derivative dy/dx. y = tan πx
> Find the derivative of the function. y = sin (tan 2x)
> Find the derivative of the function. y = ektan√x
> If H (θ) = θ sin θ, find H'(θ) and H"(θ).
> Find the derivative of the function. y = eu – e-u / eu + e-u
> Find the derivative of the function. y = sec2x + tan2x
> Find the derivative of the function. G (y) = (y2/y + 1)5
> Find the derivative of the function. y = (x2 + 1/x2 - 1)3
> Find the derivative of the function. y = 101-x2
> Find the derivative of the function. y = ex cos x
> On page 431 of Physics: Calculus, 2d ed., by Eugene Hecht (Pacific Grove, CA, 2000), in the course of deriving the formula T = 2π√L/g for the period of a pendulum of length L, the author obtains the equation aT = -g sin θ for the tangential acceleration
> Find the derivative of the function. h (t) = (t4 – 1)3 (t3 + 1)4
> Find the derivative of the function. y = (2x – 5)4 (8x2 – 5)-3
> Find the derivative of the function. y = e-2t cos 4t
> Find the derivative of the function. y = xe-kx
> Find the derivative of the function. y = 3 cot (nθ)
> Find the derivative of the function. h (t) = t3 - 3t
> Find the derivative of the function. y = a3 + cos3x
> Find the derivative of the function. y = cos (a3 + x3)
> Find the derivative of the function. f (t) = 3√1 + tan t
> Find the derivative of the function. f (z) = 1/z2 + 1
> Find equations of the tangent line and normal line to the given curve at the specified point. y = 2xex, (0, 0)
> Find the derivative of the function. f (x) = (1 + x4)2/3
> Write the composite function in the form f (g (x)). [Identify the inner function u = g (x) and the outer function y = f (u).] Then find the derivative dy/dx.
> Make a careful sketch of the graph of f and below it sketch the graph of f' in the same manner as in Exercises 4–11. Can you guess a formula for f' (x) from its graph? f (x) = sin x
> The graph shows how the average age of first marriage of Japanese men varied in the last half of the 20th century. Sketch the graph of the derivative function M' (t). During which years was the derivative negative?
> Differentiate the function. B (x) = cy-6
> Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. 6. if limx→ 6 [f (x) g (x)] exists, then the limit must be f (6) g (6). 7. I
> Determine whether the statement is true or false. If it is true, explain why If it is false, explain why or give an example that disproves the statement. 1. If f and g are differentiable, then 2. If f and g are differentiable, then 3. If f and g are
> Differentiate. y = x/2 – tan x
> Differentiate the function. h (x) = (x – 2) (2x + 3)
> Use the given graph to estimate the value of each derivative. Then sketch the graph of f'. (a). f'(-3) (b). f'(-2) (c). f'(-1) (d). f'(0) (e). f'(1) (f). f'(2) (g). f'(3)
> If R denotes the reaction of the body to some stimulus of strength x, the sensitivity is defined to be the rate of change of the reaction with respect to x. A particular example is that when the brightness x of a light source is increased, the eye reacts
