Question

Activity 1. Suppose that you have a rectangle with height h cm, base b cm, and the height is growing at a rate of 10 cm/s, and the base is growing at a rate of 20 cm/s. What is the rate that the diagonal D of the rectangle is growing when b=4 cm and h=3 cm ? Activity 2. Find the absolute max and absolute min of f(x)=1+x2x​ on the interval [−2,[infinity]).

Answer 1

1. The rate at which the diagonal
`\( D \)` of the rectangle is growing when
`\( b = 4 \)` cm and
`\( h = 3 \) cm is \( 2√(29) \) cm/s.`

2. The absolute maximum and minimum of
`\( f(x) = (1 + x^2)/(x) \)` on the interval
`\([-2, \infty)\) are \( x = -√(2) \) and \( x = √(2) \)`respectively.

Step-by-step explanation:

1. To find the rate at which the diagonal
`\( D \)` is growing, we can use the Pythagorean theorem, which relates the sides of a right-angled triangle. The diagonal
`\( D \)` is the hypotenuse of a right triangle formed by the rectangle's height
`\( h \), base \( b \),` and diagonal
`\( D \)`.

Applying the Pythagorean theorem
`\( D^2 = b^2 + h^2 \),` we differentiate both sides with respect to time
`\( t \) to obtain \( 2D (dD)/(dt) = 2b(db)/(dt) + 2h(dh)/(dt) \).` Plugging in the given values
`\( b = 4 \), \( h = 3 \), \( (db)/(dt) = 20 \) cm/s, and \( (dh)/(dt) = 10 \) cm/s,` we find
`\( (dD)/(dt) = 2√(29) \) cm/s.`

2. For finding the absolute maximum and minimum of
`\( f(x) = (1 + x^2)/(x) \)` on the interval
`\([-2, \infty)\),` we first find the critical points by setting
`\( f'(x) = 0 \).` Calculating
`\( f'(x) \)` and solving for
`\( x \)` yields critical points at
`\( x = -√(2) \) and \( x = √(2) \).`

Evaluating
`\( f(x) \)` at these points and considering the behavior as
`\( x \)` approaches infinity, we determine that
`\( x = -√(2) \)` is the absolute minimum, and
`\( x = √(2) \)` is the absolute maximum.