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Log/Exponent
Rules
Interpretation
100
The slope formula
What is m = [y1 - y2] / [x1 - x2]
100
A rule that takes certain numbers as inputs and assigns to each a definite output number.
What is a function
100
Increasing and Decreasing Function
Increasing: What is as x increases, y increases Decreasing: What is x increases, y decreases
100
Write log(x)=c in exponent form
What is 10^c=x
100
Suppose that the volume of water in a tank for 0
What is at t=2 minutes, the volume of the water is increasing at a rate of 1 Liter per minute
200
Average Rate of Change for function f(x) between x=a and x=b
What is [f(a) - f(b)] / [a - b]
200
Domain and Range of a function
Domain: What is the set of all input numbers (independent variables) Range: What is the set of resulting output numbers (dependent variables)
200
Linear Function
What is y=mx+b; a function with a constant slope
200
Simplify the following (without a calculator): ln(a)+ln(b)
What is ln(ab)
200
The elevation (ft) of a person hiking up a mountain increases at a rate of f(x)=x^2+5x (x is in hours). What does f(a)=126 mean? Could you find a?
What is when x=a, the person's elevation is increasing at a rate of 126 ft per hour. You could plug in 126 for f(a) and solve for a to get a=9 hours.
300
Estimate of the Instantaneous Rate of Change of function g(x) when x=t
What is [g(t+.001) - g(t)] / [(t+.001) - t] OR the limit of the average rates of change of g(t) over shorter and shorter intervals around t
300
Secant Line
What is a line that intersects the graph of f(x) at exactly two points; can be used to find the average rate of change between the two points of intersection
300
Concave Up and Concave Down
Concave Up: What is when the slope of the function is increasing (Note: not necessarily when the slope is positive) Concave Down: What is when the slope of the function is decreasing (Note: not necessarily when the slope is negative)
300
Expand the following (without a calculator): log(.75)
What is log(3)-log(4)
300
The temperature (in degrees F) during the day from 5:00AM to 2:00PM changes at a rate of f(x)=2^0.1x where x is the number of hours after 5:00AM. f(3)=1.231. What does f(3) represent?
What is at 8:00AM, the temperature is increasing at a rate of 1.231 degrees F per hour.
400
Relative Change
What is (Change in y)/(Original y) Note: Also called "percent change"
400
Tangent Line
What is a line that intersects the graph f(x) exactly once; can be used to estimate the slope at the point of intersection
400
Exponential Function
What is a function with a constant relative rate of change; P=P(0)a^t or P=P(0)e^(kt) Note: a=1+r
400
Simplify the following: ln(a^b)
What is b*ln(a)
400
A person is biking down a road and the function P(x) gives their position. Note, P(x) is not linear. P(5)=32 ft and P'(5)=4 ft/sec. Why is P(6)=36ft an estimate instead of an exact answer?
What is the slope of P(x) is not constant and therefore is always changing so although the P'(5)=4, P'(5.5)=??. P'(5.5) (or any other x between 5 and 6) could be bigger or smaller than 4 and thus the biker would likely not be at exactly 36ft at 6 seconds.
500
Relative Rate of Change
What is (dy/dt)/y OR f'(a)/f(a)
500
Derivative
What is the instantaneous rate of change of f(x) at the point x written as f'(x) or (dy/dx)
500
Inverse function
What is f[g(x)]=g[f(x)]; switch the x and y and solve for y; the inputs of f(x) are the outputs of the inverse: g(x) and the inputs of g(x) are the outputs of the original function: f(x)
500
Solve and explain: ln(e^x)
What is x because the exponent can be moved to the front as a multiplier and ln(e)=1 because e^1=e
500
If you're given a function that tells you the position of a car driving down a road, what would the derivative of that function tell you?
What is the velocity of the car Note: Velocity is speed with direction