Find the first four nonzero terms in the Maclaurin series for the functions in Exercises 31–38. 31. ex sin x 32. cos2 x sin x 33. cos (ex - 1) 34. ln (1 + x)/( 1 - x) 35. e^sin x 36. (tan-1 x)2 37. sin (tan-1 x) 38. cos sqrt(x) + ln (cos x)

Q: Use Taylor’s formula with a = 0 and n = 3 to find the standard cubic approximation of ƒ(x) = 1/(1 - ...

A: The given function is,

Q: solve the differential equation. yy′ = sec y2 sec 2 x

A: We need to solve the following differential equation

Q: consider the equation f(x) = 2x2 - x Find the net change between x = 3 and x = 6

A: To determine the net change of the function on the given interval.

Q: How do you simplify this expression using the properties of logarithms? 11ln(8) - 11ln(1)

A: Given expression is,11 ln(8) – 11 ln(1)

Q: I need to compute each limit. The limit is ... lim x -> -3 x2 + x - 6 ————— x3 + 27

A: Given,

Q: Consider the equation dy/dt = f(y) and suppose that y1 is a critical point—that is, f(y1) = 0. Show ...

A: To show that the equilibrium solution is asymptoticallyPart (a) Stable if f’(y1) < 0Part (b) Unst...

Q: Hi, I have some homework for my calculus class that I can't figure out. I worked it out and got it w...

A: Given,

Q: Find the lengths of the curves in Exercises 55 and 56. 55. y = (x2/8) - ln x, 4 <=x<=8 56. x =...

A: Given:

Q: The charcoal from a tree killed in the volcanic eruption that formed Crater Lake in Oregon contained...

A: The half-life of carbon-14 is 5700 years. The decay constant, k is defined to be

Q: Find the exact length of the curve. 글 cosh(2x), sxs8 y = 4 +

A: Given:

Q: 50. A tow truck drags a stalled car along a road. The chain makes an angle of 30° with the road and ...

A: According to the given information:Force = 1500 NChain makes an angle 300It is required to calculate...

Q: Find the volume of the solid generated by revolving the region bounded by the x-axis, the curve y = ...

A: Hi, thank you for the question. There are multiple question in the request. And the solution of each...

Q: You plan to store mercury (w = 849 lb/ft3) in a vertical rectangu-lar tank with a 1 ft square base s...

A: known fact:

Q: sin(2x) dx+cos(3y) dy=0,y(π/2)=π/3

A: Consider the provided differential equation with given initial condition,

Q: Find the volume of the solid generated by revolving the region in the first quadrant bounded by the ...

A: Recall the fact.

Q: Find the most general antiderivative of f(x) = -3e" + 10 sec2 (x), where - < x < 2 Note: Any arbitra...

A: According to the given information it is required to calculate the antiderivative of the function:

Q: A particle is traveling upward and to the right along the curve y = ln x. Its x-coordinate is increa...

A: Given:

Q: The estimate sqrt(1 + x) = 1 + (x/2) is used when x is small. Estimate the error when | x|&lt; 0.01.

A: Taylor inequality state that, M is a positive real number such that

Q: Find Power Series representations of the following functions (a) f(x) = 3x12 – æ5 f(x) = In(x +1) (b...

A: (a)Given

Q: For the function f,Construct and simplify the difference quotient f(x+h)-f(x)/h

A: Given:

Q: y′+(tan t)y=sin t,y(π)=0

A:

Q: lim x → 0− g(x)   lim x → 0+ g(x) Is g continuous at x = 0?

A: On the left side of x=0, y is approaching -3 So, limit x-&gt; 0^- g(x)= - 3

Q: (3x2 − 2xy + 2) + (6y2 − x2 + 3)y′ = 0

A: Given that(3x2 − 2xy + 2) + (6y2 − x2 + 3)y′ = 0(3x2 − 2xy + 2) + (6y2 − x2 + 3)(dy/dx) = 0(3x2 − 2x...

Q: A flat vertical gate in the face of a dam is shaped like the parabolic region between the curve y = ...

A:

Q: In Exercises 13–18, find the derivative of y with respect to the appro-priate variable. 13. y = 6 si...

A: We’ll answer the first question since the exact one wasn’t specified. Please submit a new question s...

Q: find the derivative (dy/dx) of the given function:  1. y= e^X sin X 2. y= (cos x)^9 3. 1/ ln(X)  4. ...

A: Given,

Q: Let ƒ be a function that is differentiable on [a, b] .  we defined the average rate of change of ƒ o...

A: Yes, this is the case. i.e. ƒ(b) - ƒ(a)/( b - a) is the  average value of ƒ′ on [a, b].The reasoning...

Q: Solve the initial value problem dy/dt+ 2y = 3, y(0) = 1

A: Given,

Q: A tank contains 100 gal of fresh water. A solution containing 1 lb/ gal of soluble lawn fertilizer r...

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Q: Suppose that q = f(p) is the demand curve for a product, where p is the selling price in dollars and...

A: Given:

Q: y′=−4t/y,y(0)=y0

A: Plug y'= dy/dtThen separate the variables.

Q: The equation x 2= 2 x has three solutions: x = 2, x = 4, and one other. Estimate the third solution ...

A: Given:

Q: 6.4.41 Question Help Let R be the region bounded by y =x², x= 1, and y= 0. Use the shell method to f...

A:

Q: Determine whether the sequences are increasing, decreasing, or not monotonic. If increasing, enter 1...

A: Find the sequence increasing, or decreasing or not monotonic.

Q: Find the volume of the solid formed by revolving the region bounded by the graphs of y = x and y = x...

A: Given information:

Q: In the function:  f(x)= (9x^2)/ ((x^2)-49) Indicate the domain and range in interval notation.  What...

A: Denominator of the function can not be equal to 0. 

Q: Consider the function f(x) whose second derivative is f"(x) = 6x + 3 sin(x). If f(0) = 4 and f'(0) =...

A: Given,

Q: In the function:  f(x)= (3x^2)ln(x) , x&gt;0 When is the function concave up and concave down in int...

A: For concavity we need second derivative.

Q: How are the disk and washer methods for calculating volumes derived from the method of slicing? Give...

A: The volume by disks for rotation about the x-axis:

Q: What is the goal of the method of partial fractions?

A: The goal of the method of partial fractions is to integrate rational functions (a ratio of polynomia...

Q: y′=(3x2−ex)/(2y−5),y(0)

A:

Q: y′=2x/(1+2y),y(2)=0

A: To determine the solution of the initial value problem and determine its maximum value.

Q: sin(2x) dx+cos(3y) dy=0,y(π/2)=π/3

A: Integrate both sides.

Q: I just need #20

A: Given,

Q: find the volume . The base of the solid is the region bounded by the parabola y2 = 4x and the line x...

A: The base of the solid bounded by the parabola y^2=4x and the line x=1. And each cross section perpen...

Q: What is the formula for integration by parts? Where does it come from? Why might you want to use it?

A: To explain integration by parts with derivation and use.

Q: Use the given transformation to evaluate the integral. lr a?dA, where R is the region bounded by the...

A: Given,

Q: How would I evaluate this indefinite integral?

A: Consider the provided indefinite integral,Simplify the given indefinite integral as follows,

Q: Consider the initial value problem y′+12y=2 cos t,y(0)=−1. Find the coordinates of the first local m...

A: Given: