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Calculus - Derivatives, Tangents and Rate of Change

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1. Differentiate the given function. Simplify your answer.

2. Differentiate the given function. Simplify your answer.

3. Differentiate the given function. Simplify your answer.

4. Find the equation of the line that is tangent to the graph of the given function at the specified point.

5. REFRIGERATION. An ice block used for refrigeration is modeled as a cube of side s. The block currently has volume of 125,000 cm3, and is melting at the rate of 1,000 cm3 per hour.
a. What is the current length s of the cube. At what rate is s currently changing with respect to time t ?
6. SPEED OF A LIZARD Herpetologists have proposed using the formula to estimate the maximum sprinting speed s (meters per second) of a lizard of mass w (grams). At what rate is the maximum sprinting speed of an 11-gram lizard increasing if the lizard is growing at the rate of 0.02 grams per day.

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Cost and Revenue Calculations using Calculus

1. The cost and the revenue functions (in dollars) for a frozen yogurt shop are given by:
C(x)= 400x +400/ x +4 and
Where x is measured in hundreds of units
A=Graph C(x) and R(x) on the same set of axes
B=What is the break-even point for this shop
C=If the profit function is given by P(x), does P(1) represent a profit or a loss ?
D= Does P(4) represent a profit or a loss?

2. The average cost (in dollars) per item of manufacturing x thousand cans of spray paint is given by:
A(x) = - .000006x^4 +.0017x^3 + .03 x^2- 24x + 1110
How many cans should be manufactured if the average cost is to be as low as possible? What is the average cost in that case?

3. Use the definition of the derivative to find the derivative of the function
Y=x^3 +5

4. Find the slope of the Tangent line to the given curve at the given value of x. Find the equation of each tangent line
Y=8-x^2; at x=1

5. Find the derivative of each of the given function
g(t)=t^3 + t-2 /(2t - 1)^5

The total energy Consumption in Quadrillion BTU for the US can be approximately by the function
f(x)=-0.00144x^3+ 0.014151x^2 +0.1388x + 23.35
Where x=0 corresponds to the year 1970
Find the energy consumption for 1990, 2000 and 2008
Find the average rate of change in energy consumption between 2000 and 2008
At what rate was energy Consumption changing in 2008

6. The Net Revenue for Chase Bank can be approximated by the function
g(x)= - .033741x^4 +1.62176x^3 - 28.4297x^2 +216.603x- 599.806 (9 <_( less than or = to) x less than or equal to 18)
Find the revenue in 2006
Find the revenue in 2008
Find the rate of change of revenue in 2007

7. Find the absolute extrema of each function on the given interval
f(x)=x^4 -18x^2 +1;[-4,4]

8. A restaurant has an annual demand for $900 bottles of a California wine. It costs $1 to store one bottle for 1 year and it costs $5 to place a reorder. Find the number orders that should be placed annually.

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