Calculus Answers

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

Show that the length of the portion of any tangent line to the asteroid a𝑥^2/3 + 𝑦^2/3 = 𝑎^2/3 ,cut off by the coordinate axes is constant.

A cone of radius 𝑟 centimeters and height ℎ centimeters is lowered point first at

a rate of 1 cm/s into a tall cylinder of radius 𝑅 centimeters that is partially filled with

water. How fast is the water level rising at the instant the cone is completely

submerged?

Suppose 𝑓 is odd and differentiable everywhere. Prove that for every positive

number 𝑏, there exists a number 𝑐 in (−𝑏, 𝑏) such that 𝑓 ′(𝑐) = 𝑓(𝑏)/𝑏.

At which point on the following curve does the tangent line has the largest slope?

𝑦 = 1 + 40𝑥^3 − 3𝑥^5