A
winch on a motionless truck 6 feet above the ground is dragging a
heavy load as indicated below. If the winch pulls the cable at a
constant rate of 1.5 ft / second, how quickly is the load moving on
the ground when it is 11 feet from the truck?Topics
Chain Rule - just a few "interesting" problems to
differentiate.
Implicit Differentiation - including find slope and equation
of tangent line at a point.
Curve Sketching without and with calculus
Related Rates
Here's how you get paid for Related Rate Problems 2 points
each
Draw a picture and Label picture (if appropriate)
Describe all variables involved and state clearly what you are trying to find.
Find equation relating the variables
Differentiate w.r.t. time
Substitute in known values to solve for desired quantity
Max and Min Problems - Given Equations, find critical values
and determine if they correspond to max or min points.
Optimization Problems - Given the situation, develop the
equation, then find max or min required.
Here's how you get paid for Optimization Problems 2 points
each
Draw a picture and Label picture (if appropriate)
Describe all variables involved and state clearly what you are trying to find.
Find equation(s) relating the variables, and write the objective variable (the one to be optimized) as a function of a single variable
Differentiate w.r.t. that single variable, find critical values, show your answer is max or min as desired.
Answer the question
Sample Problems
Chain Rule Differentiate:
1. y = sin2 4x
2. y = tan(cos 3x2)
3. y = 1 / (x2 - 3x)2
4. y = (x4(tan (3/x) )
Implicit Differentiation
1. Find dy / dx of sin3 (cos xy) = 5
2. Find dx / dy of x2 + 3x2 y - xy2 = 4
4. Find the equation of the tangent line to
(x+y)3 = x - y = 0 at the point (3, -1).
Curve Sketching: Find x & y intercepts, horizontal and vertical asymptotes and sketch
1. y = x(x-3)(x+4)2
2. y = (x-2)2(x+1)/(x+2)2 (x
-1)2
3. x2 / (x2 + 1)
Related Rates
1. A rock is thrown into a pond, causing expanding rings with radius
expanding at a constant rate of dr/dt = 6 m/s. How fast is the area
of the of the circle increasing when the radius is 20 m ?
2. A
winch on a motionless truck 6 feet above the ground is dragging a
heavy load as indicated below. If the winch pulls the cable at a
constant rate of 1.5 ft / second, how quickly is the load moving on
the ground when it is 11 feet from the truck?
3. A triangle is expanding. Find the rate of change of area when the area is 30 square inches and the height is decreasing at a rate of 3 inches / second if the base remains fixed at 6 inches.
Optimization Problems
1. Profits in the Airdale business obeys the formula P =
-A3 + 15A2 -48A - 6. How many should I
raise?
2. A farmer is building a chicken pen in the shape of a rectangle and
has 100 feet of fencing material. If one side is to be along side a
raging river, (and chickens are smart enough to know they can't
swim), what is the maximum area she can enclose?
3. What is the maximum volume of a closed cylinder with a surface area of 150 square inches?
4. A rectangular flat sheet of metal 18cm by 14 cm is to be formed
into a box by cutting out equal sized squares from each corner. What
is the maximum volume of this box?