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mathematics
calculus early transcendentals

Questions and Answers of Calculus Early Transcendentals

Use implicit differentiation to find dy/dx.exy = 2y
A spherical balloon is inflated and its volume increases at a rate of 15 in3/min. What is the rate of change of its radius when the radius is 10 in?
Evaluate the derivatives of the following functions.f(x) = tan-1 10x
Sketch a graph of g' for the function g shown in the figure. х 6. у %3D3(х) 2.
Find the following derivatives.d/dx (ln |sin x|)
Use implicit differentiation to find dy/dx.sin xy = x + y
The volume of a cube decreases at a rate of 0.5 ft3/min. What is the rate of change of the side length when the side lengths are 12 ft?
Evaluate the derivatives of the following functions.f(x) = sin-1 (esin x)
Sketch a graph of f' for the function f shown in the figure. y, y = f(x) х -2
Find the following derivatives.d/dx (ln 2x8)
Carry out the following steps.a. Use implicit differentiation to find dy/dx.b. Find the slope of the curve at the given point.tan xy = x + y; (0, 0)
A circle has an initial radius of 50 ft when the radius begins decreasing at a rate of 2 ft/min. What is the rate of change of the area at the instant the radius is 10 ft?
Use the definition of the derivative to do the following.Verify that where g(x) = √2x - 3. g'(x) V2r – 3'
Find the following derivatives.d/dx (ln x2)
Carry out the following steps.a. Use implicit differentiation to find dy/dx.b. Find the slope of the curve at the given point.cos y = x; (0, π/2)
The edges of a cube increase at a rate of 2 cm/s. How fast is the volume changing when the length of each edge is 50 cm?
Evaluate the derivatives of the following functions.f(x) = sin-1 (e-2x)
Evaluate the derivatives of the following functions.f(x) = sin-1 (ln x)
Use the definition of the derivative to do the following.Verify that f'(x) = 4x - 3, where f(x) = 2x2 - 3x + 1.
Find the following derivatives.d/dx (x2 ln x)
Carry out the following steps.a. Use implicit differentiation to find dy/dx.b. Find the slope of the curve at the given point.√x - 2√y = 0; (4, 1)
The area of a circle increases at a rate of 1 cm2/s.a. How fast is the radius changing when the radius is 2 cm?b. How fast is the radius changing when the circumference is 2 cm?
Evaluate the derivatives of the following functions.f(w) = cos (sin-1 2w)
Assume the graph represents the distance (in m) fallen by a skydiver t seconds after jumping out of a plane. a. Estimate the velocity of the skydiver at t = 15.b. Estimate the velocity of the
Find the following derivatives.d/dx (ln 7x)
Carry out the following steps.a. Use implicit differentiation to find dy/dx.b. Find the slope of the curve at the given point.sin y = 5x4 - 5; (1, π)
The hypotenuse of an isosceles right triangle decreases in length at a rate of 4/s.a. At what rate is the area of the triangle changing when the legs are 5 m long?b. At what rate are the lengths of
Evaluate the derivatives of the following functions.f(x) = x sin-1 x
Suppose the following graph represents the number of bacteria in a culture t hours after the start of an experiment.a. At approximately what time is the instantaneous growth rate the greatest, for 0
Explain the general procedure of logarithmic differentiation.
Carry out the following steps.a. Use implicit differentiation to find dy/dx.b. Find the slope of the curve at the given point.y2 + 3x = 8; (1, √5)
The legs of an isosceles right triangle increase in length at a rate of 2 m/s.a. At what rate is the area of the triangle changing when the legs are 2 m long?b. At what rate is the area of the
Evaluate the derivatives of the following functions.f(x) = sin-1 2x
The population of the United States (in millions) by decade is given in the table, where t is the number of years after 1900. These data are plotted and fitted with a smooth curve y = p(t) in the
Express the function f(x) = g(x)h(x) in terms of the natural logarithmic and natural exponential functions (base e).
Carry out the following steps.a. Use implicit differentiation to find dy/dx.b. Find the slope of the curve at the given point.y2 = 4x; (1, 2)
The sides of a square decrease in length at a rate of 1 m/s.a. At what rate is the area of the square changing when the sides are 5 m long?b. At what rate are the lengths of the diagonals of the
Explain how to find (f -1)'(y0), given that y0 = f(x0).
Suppose the height s of an object (in m) above the ground after t seconds is approximated by the function s = -4.9t2 + 25t + 1.a. Make a table showing the average velocities of the object from time t
Explain why bx = ex ln b.
Carry out the following steps.a. Use implicit differentiation to find dy/dx.b. Find the slope of the curve at the given point.x = ey; (2, ln 2)
The sides of a square increase in length at a rate of 2 m/s.a. At what rate is the area of the square changing when the sides are 10 m long?b. At what rate is the area of the square changing when the
a. Use either definition of the derivative to determine the slope of the curve y = f(x) at the given point P.b. Find an equation of the line tangent to the curve y = f(x) at P; then graph the curve
State the derivative rule for the logarithmic function f(x) = logb x. How does it differ from the derivative formula for ln x?
Carry out the following steps.a. Use implicit differentiation to find dy/dx.b. Find the slope of the curve at the given point.x4 + y4 = 2; (1, -1)
Explain why the term related rates describes the problems of this section.
a. Use either definition of the derivative to determine the slope of the curve y = f(x) at the given point P.b. Find an equation of the line tangent to the curve y = f(x) at P; then graph the curve
State the derivative rule for the exponential function f(x) = bx. How does it differ from the derivative formula for ex?
In this section, for what values of n did we prove that d/dx (xn) = nxn-1?
a. Use either definition of the derivative to determine the slope of the curve y = f(x) at the given point P.b. Find an equation of the line tangent to the curve y = f(x) at P; then graph the curve
Show that d/dx (ln kx) = d/dx (ln x), where x > 0 and k is a positive real number.
Why are both the x-coordinate and the y-coordinate generally needed to find the slope of the tangent line at a point for an implicitly defined function?
Explain how implicit differentiation can simplify the work in a related-rates problem.
Explain the differences between computing the derivatives of functions that are defined implicitly and explicitly.
Determine whether the following statements are true and give an explanation or counterexample.a. The function f|x2 = |2x + 1| is continuous for all x; therefore, it is differentiable for all x.b. If
Sketch the graph of f(x) = ln |x| and explain how the graph shows that f'(x) = 1/x.
Use x = ey to explain why d/dx (ln x) =1/x, for x > 0.
For some equations, such as x2 + y2 = 1 or x - y2 = 0 , it is possible to solve for y and then calculate dy/dx. Even in these cases, explain why implicit differentiation is usually a more efficient
Use the General Power Rule where appropriate to find the derivative of the following functions.f(x) = xe
Second derivatives Find d2y / dx2.x4 + y4 = 64
For the situation described in Exercise 32, what is the rate of change of the area of the exposed surface of the water when the water is 5 m deep?Data from Exercise 32A hemispherical tank with a
Evaluate and simplify the following derivatives.f'(1) when f(x) = x1/x
Find an equation of the line tangent to the graph of f at the given point.f(x) = cos-1 x2; (1/√2,π/3)
Second derivatives Find d2y / dx2.x + y = sin y
A hemispherical tank with a radius of 10 m is filled from an inflow pipe at a rate of 3 m3/min (see figure). How fast is the water level rising when the water level is 5 m from the bottom of the
Evaluate and simplify the following derivatives.d/dx (xsin x)
Find an equation of the line tangent to the graph of f at the given point.f(x) = sin-1 (x/4); (2, π/6)
The energy (in joules) released by an earthquake of magnitude M is given by the equation E = 25,000 • 101.5M. (This equation can be solved for M to define the magnitude of a given earthquake; it is
Second derivatives Find d2y / dx2.2x2 + y2 = 4
Water is drained out of an inverted cone having the same dimensions as the cone depicted in Exercise 29. If the water level drops at 1 ft/min, at what rate is water (in ft3/min) draining from the
Find an equation of the line tangent to the graph of f at the given point.f(x) = tan-1 2x; (1/2, π/4)
The following table shows the time of useful consciousness at various altitudes in the situation where a pressurized airplane suddenly loses pressure. The change in pressure drastically reduces
Second derivatives Find d2y / dx2.x + y2 = 1
At what rate is soda being sucked out of a cylindrical glass that is 6 in tall and has a radius of 2 in? The depth of the soda decreases at a constant rate of 0.25 in/s.
Evaluate and simplify the following derivatives. sin dx
Evaluate and simplify the following derivatives.d/dx (log3 (x + 8))
Evaluate the derivatives of the following functions.f(x) = 1/tan-1 (x2 + 4)
Find the derivatives of the following functions.y = ln 10x
Carry out the following steps.a. Verify that the given point lies on the curve.b. Determine an equation of the line tangent to the curve at the given point.(x2 + y2)2 = 25 / 4 xy2; (1, 2) УА 1 х 2
An inverted conical water tank with a height of 12 ft and a radius of 6 ft is drained through a hole in the vertex at a rate of 2 ft3/s (see figure). What is the rate of change of the water depth
Evaluate and simplify the following derivatives.d/dx (2x2 - x)
Evaluate the derivatives of the following functions.f(s) = cot-1 (es)
Find the derivatives of the following functions.A = 250(1.045)4t
Carry out the following steps.a. Verify that the given point lies on the curve.b. Determine an equation of the line tangent to the curve at the given point.cos (x - y) + sin y = √2; (π/2, π/4)
A water heater that has the shape of a right cylindrical tank with a radius of 1 ft and a height of 4 ft is being drained. How fast is water draining out of the tank (in ft3/min) if the water level
Evaluate the derivatives of the following functions.f(x) = sin (tan-1 (ln x))
Evaluate and simplify the following derivatives.d/dw (e-w ln w)
Find the derivatives of the following functions.P = 40/1 + 2-t
Carry out the following steps.a. Verify that the given point lies on the curve.b. Determine an equation of the line tangent to the curve at the given point.x3 + y3 = 2xy; (1, 1) y
Sand falls from an overhead bin and accumulates in a conical pile with a radius that is always three times its height. Suppose the height of the pile increases at a rate of 2 cm/s when the pile is 12
Evaluate and simplify the following derivatives.d/dx (x ln2 x)
Evaluate the derivatives of the following functions.f(x) = csc-1 (tan ex)
Find the derivatives of the following functions.y = x3 • 3x
Carry out the following steps.a. Verify that the given point lies on the curve.b. Determine an equation of the line tangent to the curve at the given point.sin y + 5x = y2; (π2 / 5 , π) yA 5
Runners stand at first and second base in a baseball game. At the moment a ball is hit, the runner at first base runs to second base at 18 ft/s; simultaneously, the runner on second runs to third
Evaluate and simplify the following derivatives.d/dx (xe-10x)
Evaluate the derivatives of the following functions.f(x) = tan-1 (e4x)
Find the derivatives of the following functions.y = 4-x sin x
Carry out the following steps.a. Verify that the given point lies on the curve.b. Determine an equation of the line tangent to the curve at the given point.x4 - x2y + y4 = 1; (-1, 1) х
A 5-foot-tall woman walks at 8 ft/s toward a streetlight that is 20 ft above the ground. What is the rate of change of the length of her shadow when she is 15 ft from the streetlight? At what rate is

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