In this section we prepare for the final exam.
1 Limits
2 Derivatives
Find the equation of the tangent line to the graph of at , and use it to approximate
.
The equation is and .
The equation is and .
3 Integrals
4 FTC, Part II
5 Implicit Differentiation
6 Related Rates
The radius of a circle is increasing at a rate of cm/min. How fast is the area
increasing when the radius is cm?
The radius of a sphere is increasing at a rate of cm/min. How fast is the volume
increasing when the radius is cm?
The short leg of a right triangle is increasing at a rate of cm/min and the long leg is
increasing at a rate of cm/min. How fast is the hypotenuse increasing when the short
leg is cm and the long leg is cm?
7 Optimization
A farmer has 1200 feet of fence with which to fence in a rectangular pasture.
Moreover, he would like to partition the pasture into two plots with fence parallel to
the bottom side (horizontal). What dimensions will maximize the total area of his
enclosed pastures?
The optimal length (horizontal) is feet and
the optimal width (vertical) is feet.
The optimal length (horizontal) is feet and
the optimal width (vertical) is feet.
A fifth grader has cm of cardboard with which to construct a box with a
square base and an open top. What is the maximum possible volume for his
box?
The maximum possible volume is .
The maximum possible volume is .
8 Max/Min, Inc/Dec and Concavity
Find the absolute maximum and the absolute minimum of the function on the
interval .
The absolute maximum is occurring at .
The absolute minimum is occurring at .
The absolute maximum is occurring at .
The absolute minimum is occurring at .
Find the absolute maximum and the absolute minimum of the function on the
interval .
The absolute maximum is occurring at .
The absolute minimum is occurring at .
The absolute maximum is occurring at .
The absolute minimum is occurring at .
9 Rectilinear Motion
A projectile is launched vertically. Its height (in feet) after seconds is given by
.
The maximum height of the projectile is .
The velocity of the object when it is feet up and falling is .
A projectile is launched vertically. Its height (in feet) after seconds is given by
.
The maximum height of the projectile is .
The velocity of the object when it is feet up and rising is .
10 Mean Value Theorem
Show that there is no value of ‘’ from the Mean Value Theorem for the
function on the interval . Why does this NOT contradict the statement of the
theorem?