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Algebra
A function f is said to have a removable discontinuity at a if:
1. f is either not defined or not continuous at a.
2. f(a) could either be defined or redefined so that the new function is continuous at a.
Let f(x)=(2x2+5x−7)/(5x−5).
Show that f has a removable discontinuity at 5 and determine the value for f(5) that would make f continuous at 5.
Need to redefine f(5)=
Algebra
Pretend you had to give quick directions to your friend right before the quiz about how to stretch functions vertically, shift them horizontally, and shift them vertically. How would you do that as simply as possible?
Algebra
Consider these statements written in ordinary language:
A The speed of the car is proportional to the distance it has traveled.
B The car is speeding up.
C The car is slowing down.
D The car always travels the same distance in the same time interval.
E We are driving backwards.
F Our acceleration is decreasing. Denoting by s(t) the distance covered by the car at time t, and letting k denote a constant, match these statements with the following mathematical statements by entering the letters A through F on the appropriate boxes:
s′′<0.
s′ is constant
s′<0
s′′′<0
s′′>0
s′=ks
A The speed of the car is proportional to the distance it has traveled.
B The car is speeding up.
C The car is slowing down.
D The car always travels the same distance in the same time interval.
E We are driving backwards.
F Our acceleration is decreasing. Denoting by s(t) the distance covered by the car at time t, and letting k denote a constant, match these statements with the following mathematical statements by entering the letters A through F on the appropriate boxes:
s′′<0.
s′ is constant
s′<0
s′′′<0
s′′>0
s′=ks
Algebra
f you toss a rock at an initial height H with an initial velocity V then its height h(t) after t seconds is given by the formula
h(t)=−12gt2+Vt+H
where
g=32feetsecond2
on Earth. (On other celestial bodies g would be different. The minus sign is due to the convention that the positive direction is up and gravity pulls down.)
You have probably seen this formula before, and chances are you were simply told that this is the way it is. With Calculus, we can make perfect sense of this formula. The underlying observation is the experimentally observed fact that, ignoring air resistance, on Earth a free falling object increases its speed by 32 feet per second every second. The velocity of the object is the derivative of height, and the acceleration is the derivative of velocity. So we have to work out that the formula given above has the right properties.
If h is given by the above expression, then
h(0)=
.
The velocity v(t) is
v(t)=h′(t)=
and
v(0)=
.
Moreover, the acceleration is
v′(t)=
.
h(t)=−12gt2+Vt+H
where
g=32feetsecond2
on Earth. (On other celestial bodies g would be different. The minus sign is due to the convention that the positive direction is up and gravity pulls down.)
You have probably seen this formula before, and chances are you were simply told that this is the way it is. With Calculus, we can make perfect sense of this formula. The underlying observation is the experimentally observed fact that, ignoring air resistance, on Earth a free falling object increases its speed by 32 feet per second every second. The velocity of the object is the derivative of height, and the acceleration is the derivative of velocity. So we have to work out that the formula given above has the right properties.
If h is given by the above expression, then
h(0)=
.
The velocity v(t) is
v(t)=h′(t)=
and
v(0)=
.
Moreover, the acceleration is
v′(t)=
.
Algebra
You toss a rock up vertically at an initial speed of 40 feet per second and release it at an initial height of 6 feet. The rock will remain in the air for how many
seconds.
It will reach a maximum height of how many
feet after how many
seconds.
Note: Ignore air resistance.
seconds.
It will reach a maximum height of how many
feet after how many
seconds.
Note: Ignore air resistance.
Algebra
Consider the function f(x)=−2x2+6x−9. f(x) is increasing on the interval (−∞,A] and decreasing on the interval [A,∞) where A is the critical number.
Find A
At x=A, does f(x) have a local min, a local max, or neither? You should be aware that there are two different tests that may work. Either use the Second Derivative test and look at concavity at A, or use the First Derivative test and examine the sign table.
Type in your answer as LMIN, LMAX, or NEITHER.
Find A
At x=A, does f(x) have a local min, a local max, or neither? You should be aware that there are two different tests that may work. Either use the Second Derivative test and look at concavity at A, or use the First Derivative test and examine the sign table.
Type in your answer as LMIN, LMAX, or NEITHER.
Algebra
The function f(x)=2x3−24x2+72x−6 has two critical values. The smaller one equals
and the larger one equals
and the larger one equals
Algebra
The function f(x) = 5-5x^4 has an absolute value of what and this occurs at x equals
Algebra
Differentiate the following function: f(x)=e^x+x^e
Algebra
How would you convince a child that any number multiplied by 0 is 0, using a
teaching aid.
teaching aid.
