Big Ideas
How do you use v(t) to find when an object is "at rest"
v(t)=0
Given s(t)=4t^3-2t^2+5t , find an expression for a(t)
a(t)=24t-4
Given v(t)=6e^(-.5t+3) , find the initial velocity
v(0)=121
What does "Initially" mean
t=0
Given f(x)=x^2-5 and g(x)=x/2 , find g(f(5))
g(f(5))=10
How do you use s(t) to find when an object is "changing directions"
Find maximums/minimums
Given v(t)=2t^2-32 , find when the particle is at rest for t>=0
t=4
Given s(t)=sin(t^2-5) , find the velocity at t=2
v(2)=2.16
The function d(x) tells researchers the number of deer in a protected forest x months after they began recording. What does d'(5)=-100 represent in units?
After 5 months, the deer population was decreasing by 100 deer per month
Given f(x)=(x^2+4)/3 , write an expression for f^-1(x)
f^-1(x)=sqrt(3x-4)
How can you tell if an object is speeding up?
v(t) and a(t) have the same sign
Given v(t)=t^2-4t , determine if the object is slowing down or speeding up at t=3 , giving a reason
v(3)=-3 a(3)=2
Since v(3) and a(3) have different signs, it is slowing down
Given v(t)=t^3-1 for 0<=t<=4 , find the total distance travelled by the particle
61.5
The function f(t)=2sin(pi/6t) models deviation in ocean temperature throughout the year where t is the number of months after April. Find the rate at which the temperature is changing in May
f(1)=0.907 (degrees per month)
Find the values of k for which y=kx^2+2x-5 has no real roots
k<-4/20 or k<-1/5
How do you calculate total distance from t_1 to t_2 using v(t) ?
int_(t_1)^(t_2)|v(t)|dt
Given a(t)=t^2+8t-5 and an initial velocity of 7, find an expression for v(t)
v(t)=1/3t^3+4t^2-5t+7
Given s(t)=e^((x^2-3x)/(2x^3+1)) for 0<=t<=5 , find the total distance travelled by the particle
1.31
The function f(t)=2sin(pi/6t) models deviation in ocean temperature throughout the year where t is the number of months after April. During which month is the temperature decreasing the fastest?
October (t=6)
A quadratic with a vertex at (-2, 6) and a y-intercept at (0,3) can be written in the form y=a(x-h)^2+k . Find the values of h , k , and a
h=-2
k=6
a=-3/4
How do you calculate total displacement from t_1 to t_2 given only v(t) ?
int_(t_1)^(t_2)v(t)dt
Given s(t)=sin(5t)+3cos(t) , find an expression for a(t)
a(t)=-25sin(5t)-3cos(t)
Given s(t)=t^3-5t^2-t-1 for t>=0 , write the interval over which the particle is slowing down
(1.67, 3.43)
or
1.67<t<3.43
The function f(x)=4e^(-.1x)+30 models the number of atoms (in millions) in a chemical sample x years after it was produced. Find the limit of f'(x) as x approaches infinity
0
Given the position function: s(t)=2t^4-36t^2 for t>=0 , algebraically determine any values of t when the object is at rest
t=0
t=3