Calculus 101 Midterm 2
Review List, Prof. Donnay
You will be allowed one side of a 3 x 5 index card on which you can write notes for the
midterm. As part of what you write, you can include formulas for area and volume that
you might need in a related rates problem (ex. Volume of sphere, cylinder, cone. )
The following are the topics we have covered and that could be on the test:
S3.3: Differentiation formulas: Product, quotient.
S.3.5: Chain Rule:
Take the derivative of the following functions. You do not have to simplify.
f(x) = (3x5 – 4x2 + 2x -5)8
f(x) = (4x2+3x)(9x7-6x4+3x)
f(x) =(4x2+3x) / (9x7-6x4+3x )
f(x) = (4x2+3x)5(9x7-6x4+3x)3
f(x) =(4x2+3x)3 / (9x7-6x4+3x )2
f(x) = ((4x2+3x) / (9x7-6x4+3x ))3
S3.4: Trig review, Derivatives of Trig functions.
- you should know by heart the derivatives of sin(x), cos(x), tan(x). With this
derivatives, you should be able to derive the formula for the derivative of sec(x), csc(x),
cot(x).
Take the derivative of: f(x) = sin(3x)
f(x) = cos2(7x)
f(x) = sin( 2x) cos( 3x)
f(x) = tan(x2)
1
f(x) = sin(x) + cot(x)
2
Summary: you should be able to take the derivative of a variety of formulas and functions
! several rules might be used in one problem.
in which
3.7: Rates of change in physical problems.
Water is flowing out of tank. The volume of water in the tub is given by
V(t) = 100( 1 – t/30 )2 , 0 " t " 30 , where the volume is in gallons and the time is in
minutes. What is the rate at which the water is draining from the tank at t=10 minutes?
Give units.
! (This is an application of the Chain Rule).
3.8: Related rates.
A puddle’s radius is expanding at 3 cm/sec. What is the area’s rate of expansion when the
radius is 6 cm?
The area of a circular pool of water is expanding at a rate of 25 " in2/sec. At what rate is
the radius expanding when the radius is 4 inches?
A spherical ball has its diameter increasing at a rate of 2 inches/sec. How fast is the
!
volume of the ball changing when the radius is 10 inches?
A woman and a man are walking in the same direction but are 2 ft apart. They stop and
the man begins walking in the other direction at 5 ft/sec. The woman continues in the
original direction at 3 ft/sec. After 30 secs, how fast are they moving apart?
Two ships leave port at the same time (10 am). One ship travels south at 40 miles/hr. The
other ship travels east at 80 miles/hr. How fast is the distance between the ships
increasing at 5 pm?
A 5 ft ladder is sliding down the wall at 2 ft/sec. How fast is the bottom of the ladder
sliding away from the wall when the top of the ladder is 4 ft above the ground?
3.9: Linear Approximations: be able to write the linear approximation function L(x) for a
function f(x) based at a point x0. Use the linear approximation to estimate the value of
f(x) for x near x0. There will not be problems involving differentials on this midterm.
Give the linear approximation function L(x) that approximates the function f(x) = 3x2
near x0= 2. Use the linear approximation to estimate f(2.01). What is the exact value of
f(2.1). What is the error between the exact value f(2.01) and the approximate value
L(2.01)? Draw a picture that shows both f(x) and L(x) and that shows this error.
Give the linearization L(x) of the function f(x) = 4x3 + 3x2 at a = 2. Estimate the value of
f( 1.99) by using the linearization.
S. 4.1 Max/Min of Function (Extreme value Theorem: A continuous function on a
closed interval attains its absolute max and min).
- Understand the difference between a local min/max and an absolute max/min.
- Be able to find the local and absolute max/min from a graph
- Be able to find the absolute max/min of a function on a closed interval (find
critical points, compare value of function at critical points to value of function and
endpoints).
Find the absolute max and min of the function f(x) = 2x3 – 3x2 – 12x +3 on the closed
interval [-2, 4].
S. 4.3: Graphing Functions.
- Be able to graph a function using the first and second derivatives.
-Determine when a function is increasing or decreasing.
- Find critical points of function. Determine if critical points are local max/min by
using the 1st Derivative Test or by using the 2nd Derivative Test.
- Determine when the function is concave up or concave down.
- Determine points of inflection.
- Put all this information together to give a detailed graph of a function.
Graph the function f(x) = 2x3 – 3x2 – 12x +3. Find the intervals on which f(x) is
increasing and decreasing. Find the intervals on which f(x) is concave up and concave
down. Find the local maximums and local minimums. Use the first derivative test (or the
2nd derivative test) to determine if a critical point is a local max or min.
4.2 Mean Value Theorem
- Understand what the Theorem says (I will write out the Theorem on the test if there is a
question about it) so that you can find a point c that satisfies the statement of the
Theorem.
- From a graph, be able to draw a tangent line whose slope f’(c ) matches the slope of the
secant line between ( a, f(a) ) and ( b, f(b) ).
The Mean Value Theorem says that under appropriate conditions,
f (b) " f (a)
= f '(c) for some c " [a,b] .
b"a
!
For f(x) = x3 +2x -5 and x " [#2,3] , find all points c that satisfy the Mean Value
Theorem.
!
A car is driving straight west on the highway. At 4pm, the car is 125 miles west of
! 4:15 pm, the car is 142 miles west of Philadelphia. Show that at some
Philadelphia. At
time between 4pm and 4:15 pm, the car is going exactly 68 miles/hr.