Radians
True or False
Includes a 0
Modeling
Graphs
100
The radian measure corresponding to 90 degrees.
\frac{\pi}{2}
100
One time around the unit circle is an angle of \pi radians.
FALSE! One time around the unit circle (in a counterclockwise direction) corresponds to
2 \pi
100
What is
ln(1)
0
100
In f(t)=Pe^{rt} which letter represents the continuous growth rate?
r. Remember that you need to write r as a decimal, so if the rate is 5%, write r=0.05
100
What is y=sin(x)?
200
Give two angles where the function f(\theta) =-2sin(\theta) has the same y values as
f(\theta)=\sin(\theta)
0, \pi, 2\pi, 3\pi, 4\pi, -\pi, -2\pi
200
The solution to 2=e^{x} is greater than 1.
False! 0< ln2<1
200
Find an angle on the unit circle with the same sine value as
\theta=\pi
0 or
2\pi
200
If a sinusoidal function has amplitude 3 and midline 4, which is a possible model?
(1) y=3\sin(\theta)+4
(2) y=3\cos(\theta+4)
(3) y=4sin(\theta) +3
1! The amplitude multiplies by the values of the sine or cosine. The midline is added on to each output of sine or cosine.
2 is wrong because the "+4" is inside parentheses (so it's just shifting your starting angle, not changing the midline)
3 has the midline and amplitude swapped.
200
Where on the graph of an exponential function y=a\cdot b^x can I find the value of a?
The y-intercept (y value where x=0).
300
Give the value of 2 radian angles \theta <2\pi with \cos(\theta)=\frac{-\sqrt(2)}{2}
-\frac{3\pi}{4}, -\frac{5\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}
300
To solve 40=5e^{0.3t} we need to solve the equation \ln 8=0.3t
True! First divide 40=5e^{0.3t} by 5 on both sides to get the equation below, then convert to "log form"
8=e^{0.3t}
300
The solution to the equation 5 \cdot 10^{x+1}=5
What is x=-1
300
A bacteria population of 10 cells doubles every 30 minutes.
What is formula for the population after t HOURS?
P(t)=10\cdot 2^{2t}
P(t)=10\cdot 4^t
300
Where will the graph of the function y=\log_5x cross the x-axis?
When x=1.
\log_5(x)=0
5^0=x
x=1
400
This angle \theta satisfies:
1) \theta>2\pi
2) \sin(\theta)=1
\frac{5\pi}{2}, \frac{9\pi}{2}, etc
400
If a population is growing by 4% every 6 months, it is growing by 8% every year
False! Counting in months, we know every 6 months to multiply by 1.04. So for 12 months, we multiply by 1.04 two times:
1.04\cdot 1.04=1.0816
=8.16% growth
400
The graph of the function f(x)= 50 \cdot b^x includes the point (2, 500). What is b?
b^2=10
b=\sqrt(10)
400
In an exponential model a\cdot b^x , I know a=100 and that (2,900) is on the graph of the function. What equation helps me to find b?
(1) 900=100b^2 or (2) \log_b(9)=2 ?
Definitely 1. Solve it using roots (fractional exponents).
9=b^2 \rightarrow \sqrt9=b=3
400
What is the value of b on the graph of y=a\cdotb^x shown here?
b=1/4 since from x=1 to x=2, we have b=(end)/start= 10/40=1/4
500
The angle with \sin(\theta)= \frac{\sqrt2}{2} and \tan(\theta)=-1
\theta=\frac{3\pi}{4}
500
Adding \pi to an angle is the same as subtracting \pi from an angle
Kinda both. The angle will land on the same place on the unit circle either way, but the angle will mean something different (namely, a counterclockwise v. clockwise rotation of 180 degrees). Example:
\frac{\pi}{4}+\pi=\frac{5\pi}{4}
\frac{\pi}{4}-\pi=-\frac{3\pi}{4}
500
An exponential function has the points (2,20) and (3,10).
What is f(0)?
b= \frac{10}{20}=0.5
f(0)= \frac{20}{b^2}=\frac{20}{0.5^2}
f(0)=20\cdot 4=80
500
A bank account grows via the continuous growth model Pe^{rt} with initial investment $99 and a growth rate of 4%.
How long will it take for the investment to grow to $500?
40.49 years.
ln(500/99)=4.04t
t= ln(500/99)/0.04 =40.49
500
Name this function.
One option is to see it as a reflection and stretch of cosine, so y= -2cos(x) since the amplitude is 2, and it starts at the minimum instead of the maximum at x=0.
Another is to think of it as a shift of 2sinx, moving pi/2 to the right. So y=2sin(x-pi/2) since at 0, the function output looks like the value of sinx when the angle is -pi/2.