# More derivatives!

Q:  Find the derivative when x = 1 of the function f(x) = 1/√(x)

# Average rate of change vs. Instantaneous rate of change

Q:  Let f(x) = 3x² – x

a.  Find the average rate of change of f(x) with the respect to x as x changes from x = 0 to x = 1/16

b. Use calculus to find the instantaneous rate of change of f(x) at x=0 and compare with the average rate found in part (a).

# Tangent Line

Q:  Find the derivative of function, then find the equation of the line that is tangent to its graph for the specified value x=c

f(x) = x²; c=1

# Find the equation of the tangent line!

Q:  Let f(x) = -2/x .  Find the equation of the tangent line when x = -1.

# Horizontal Tangent Line

Q:  Let f(x)= 3sin2x + 4cos3x.  Determine where the tangent line is horizontal.

# Quotient with chain rule

Q:  F(x) = (3x²+1)3/(x2-1)4.  Find the derivative of F(x), F'(x).

BACKGROUND

First we need some basics (assuming everything is linear, we continue):

We logically know that:

Profit = What you make – What you spend

In math, that is:

P = Revenue – Cost

(1) P = R – C

And,

Revenue = price * quantity

(2) R = px

(3) Cost = (variable cost)*x + (fixed cost)

Now, there is a difference between big P (profit) and little p (price or demand)

We usually assume price is linear, so:

(4) p = mx + b

Everything in BOLD are things you must know!

OK…. Now let’s start deciphering the actual problem:

Q:  A manufacturer sells 150 tables a month at the price of \$200 each. For each \$1 decrease in price, he can sell 25 more tables. The tables cost \$125 to make. Express monthly profit as a function of the price, draw a graph and estimate the optimal selling price.