Exam 4: Applications of Differentiation

There are 50 apple trees in an orchard. Each tree produces 100 lbs of apples. For each additional tree planted in the orchard, the output per tree drops by 2 lb. How many trees should be added to the existing orchard in order to maximize the total output of trees?

Find the absolute minimum value for the function $f ( x ) = 2 \sin x \cos x \text { for } 0 \leq x \leq \pi$

If Newton's method is used to solve $2 x ^ { 3 } + 2 x + 1 = 0$ with first approximation $x _ { 1 } = - 1$ , what is the second approximation, $x _ { 2 }$ ?

Find the most general antiderivative of the function $f ( x ) = \frac { 1 + 3 x } { \sqrt { x } }$ .

Which of the statements below are true? 1) If $f ^ { \prime } ( c ) = 0$ , then f(x) has a maximum or minimum value at $x = c .$ 2) If $f ^ { \prime } ( x ) = g ^ { \prime } ( x )$ for all x in an interval I, then $f ( x ) = g ( x )$ on I.3) If f(x) is differentiable on the open interval $( a , b )$ , and c is a point of local maximum for f in $( a , b )$ , then $f ^ { \prime } ( c ) = 0$ .

A farmer has 20 feet of fence, and he wishes to make from it a rectangular pen for his pig Wilbur, using a barn as one of the sides. In square feet, what is the maximum area possible for this pen?

Find the most general antiderivative of the function $f ( x ) = 2 x ^ { 3 } - x ^ { 2 } + 5$ .

What is the smallest slope for a tangent line to the graph of $y = 2 x ^ { 3 } - 2 x ^ { 2 } + 5 x + 1$ ?

Two cars start moving from the same point. One travels south at v miles per hour and the other travels west at w miles per hour. Show that the rate of the distance between two cars is changing at a constant rate of $\sqrt { v ^ { 2 } + w ^ { 2 } }$ miles per hour.

Given $f ^ { \prime \prime } ( x ) = 2 , f ^ { \prime } ( 0 ) = 1$ , and $f ( 0 ) = \frac { 3 } { 2 }$ , find $f ( 1 )$ .

A farmer intends to fence o a rectangular pen for his pig Wilbur, using the barn as one of the sides. If the enclosed area is to be 50 square feet, what is smallest amount of fence needed, in feet?

A lighthouse is on a small island 3 km away from the nearest point $P$ on a straight shoreline, and its light makes 4 revolutions per minute. At what rate is the beam of light moving along the shoreline when it is 1 km away from $P$ ?

Find the most general antiderivative of the function $f ( x ) = \frac { 2 } { x ^ { 3 } }$ .

A rectangular region with area 3200 square feet is to be enclosed within a fence. The two sides which run north-south will use fencing materials costing $1.00 per foot, while the other two sides require fencing materials which cost $2.00 per foot. Find the dimensions of the region which gives the smallest material cost.

An open box is made from a 16 inch $\times$ 16 inch piece of cardboard by cutting equal squares from each corner and folding up the sides. For maximum volume, what size squares should be cut out?

Two cars are each 100 miles away from the town of Tucumcari, one directly to the north and the other directly to the east. The car to the north is heading toward the town at 60 miles per hour, while the one to the east is heading toward the town at 30 miles per hour. How fast are the cars approaching each other?

A man who is 6 feet tall walks away from a 24 foot tall lamppost at a constant rate of w feet per second. Does the tip of the man's shadow also move at a constant rate? Explain your answer.

A particle moves along the circle $x ^ { 2 } + y ^ { 2 } = 25$ . If x increases at a rate 2 units per minute, at what rate is y changing when $x = 3$ ?

Use the graph of f to estimate the value of c that satisfy the Mean Value Theorem for the interval $[ 0,8 ]$ .

Find the value of the limit $\lim _ { x \rightarrow 0 } \left( \frac { 1 } { x - 1 } - \frac { 1 } { \ln x } \right)$ .