Q: Find the derivative when x = 1 of the function f(x) = 1/√(x)
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).
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
Q: Let f(x) = -2/x . Find the equation of the tangent line when x = -1.
Q: Let f(x)= 3sin2x + 4cos3x. Determine where the tangent line is horizontal.
Q: F(x) = (3x²+1)3/(x2-1)4. Find the derivative of F(x), F'(x).
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
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.
Q: Find the point(s) on the graph of the function f(x) = (x² + 6)(x – 5) where the slope of the tangent line is -2.