Calculus Answers

What is the Jacobian matrix J(r, θ) for the polar coordinate transformation, given that x=rcos⁡θ and y=rsin⁡θ.

Select one:

A. cos⁡θ-rsin⁡θ-sin⁡θrcos⁡θ

B. -rcos⁡θsin⁡θsin⁡θ-rcos⁡θ

C. cos⁡θ-rsin⁡θsin⁡θ-rcos⁡θ

D. cos⁡θrsin⁡θsin⁡θrcos⁡θ

A spring of natural length 10 in. stretches 1.5 in. under a weight of 8 lb. Find the work done in stretching the spring

(a) from its natural length to a length of 14 in.

(b) from a length of 11 in. to a length of 13 in.

Determine where the global extrema of  f(x)=3x^2/3−2x

on the interval [−1,1] occur.

Find the global maximum and minimum values of the following function on the given interval. If there are multiple points in a single category list the points in increasing order in x value and enter N in any blank that you don't need to use.

f(x)=4e−x−4e−2x  [0,1]

Global maxima

x = ____ Y=___

x=_____ Y=____

x=____ Y=___

Global minima

x=____ Y=___

x=____ Y=___

x=___ Y=___

x=____ Y=___

Graph the following function:

y= e^x / x²

Ensure that you include the following properties:

• Asymptote, x, y – intercepts, Domain and Range
• First derivative, Maximum and Minimum
• Second derivative, inflection points
• Regions or increasing and decreasing
• Regions or concave up and concave down
• Sketch of the graph

Y= tan^-1(x+1/2x+3)

Find the limit f(x),if it exists.sketch the graph of f(x).

Limit x approaching zero f(x),where f(x)=2-x,x less than or equal zero,. x+2, x>0

Show whether the following equation are exact and hence solve the equation.

a) 2x(ye^x² -1)dx + e^x²dy= 0

b) ( 6x⁵y³ + 4x³y⁵) dx + (2x⁶y² - 5x⁴y⁴) dy =0

A region is bounded by y = square root of x, the x-axis, and x = 4. Write the integral that represents the volume of this region revolved about the line y = 3