Calculus Answers

Find the local and absolute extreme values of the function on the given interval. Also

specify the intervals where function is increasing or decreasing

(i) 𝑓(𝑥) = (𝑥²+ 𝑥 + 1)²

Q: Find the local and absolute extreme values of the function on the given interval. Also

specify the intervals where function is increasing or decreasing

𝑓(𝑥) = 𝑥²e^-x

The equation for a displacement 𝑠(𝑚), at a time 𝑡(𝑠) by an object starting at a displacement of 𝑠0 (𝑚), with an initial velocity 𝑢(𝑚𝑠 −1 ) and uniform acceleration 𝑎(𝑚𝑠 −2 ) is: 𝑠 = 𝑠0 + 𝑢𝑡 + 1 2 𝑎𝑡 2 A projectile is launched from a cliff with 𝑠0 = 30 𝑚, 𝑢 = 55 𝑚𝑠 −1 and 𝑎 = −10 𝑚𝑠 −2 . The tasks are to: a) Plot a graph of distance (𝑠) vs time (𝑡) for the first 10s of motion. b) Determine the gradient of the graph at 𝑡 = 2𝑠 and 𝑡 = 6𝑠. c) Differentiate the equation to find the functions for: i) Velocity (𝑣 = 𝑑𝑠 𝑑𝑡) ii) Acceleration (𝑎 = 𝑑𝑣 𝑑𝑡 = 𝑑 2 𝑠 𝑑𝑡2 ) d) Use your results from part c to calculate the velocity at 𝑡 = 2𝑠 and 𝑡 = 6𝑠. e) Compare your results for part b) and part d). f) Find the turning point of the equation for the displacement 𝑠 and using the second derivative verify whether it is a maximum, minimum or point of inflection. g) Compare your results from f) with the graph you produced in a).

A delivery company accepts only rectangular boxes the sum of whose length and the perimeter of a cross-section does not exceed 108 inches. Find the dimensions of an acceptable box of largest volume. 

The edge of a cube was found to be 30 cm with a possible error in measurement of .1 cm. Use differentials to estimate the percentage error (to the nearest hundredth) in computing (a) the volume of the cube and (b) the surface area of the cube.

 An open box is to be made out of a 8-inch by 16-inch piece of cardboard by cutting out squares of equal size from the four corners and bending up the sides. Find the dimensions of the resulting box that has the largest volume.

A rectangular billboard 5 feet in height stands in a field so that its bottom is 6 feet above the ground. A nearsighted cow with eye level at 4 feet above the ground stands x

x

 feet from the billboard. Express θ

θ

, the vertical angle subtended by the billboard at her eye, in terms of x

x

. Then find the distance x

x

 the cow must stand from the billboard to maximize θ

θ

.

Solve g(y)= 1/y over the interval (1,4)

The finite region bounded by the planes z = x, x + z = 8, z = y, y = 8, and z = 0 sketch the region in R3 write the 6 order of integration. No need to evaluate. clear writing please

A rectangular billboard 6 feet in height stands in a field so that its bottom is 13 feet above the ground. A nearsighted cow with eye level at 4 feet above the ground stands x

x

 feet from the billboard. Express θ

θ

, the vertical angle subtended by the billboard at her eye, in terms of x

x

. Then find the distance x

x

 the cow must stand from the billboard to maximize θ

θ

.