Definitions
How a quantity changes relative to another. In calculus: change in y over change in x
What is Rate of Change?
Given f(x)=1/x, find the AROC from x=1 to x=3
What is -1/3?
Find the IROC of f(x)=x3+x at x=−2
What is 13?
Write the equation tangent to f(x) = sqrt(x) at x = 4
L(x)=2+41(x−4)
What determines whether linear approximation over/underestimates a value?
The concavity of the function at that point.
f(b) - f(a) / (b - a); slope of the secant line between two points.
What is Average Rate of Change?
The position of a particle is given by s(t)=2t3−9t2+12t. Find its average velocity on [1, 4].
What is 9?
Given a table:
f(1.9)=3.2, f(2.0)=3.4, f(2.1)=3.7, estimate f′(2).
What is 1.25?
Use your linearization to approximate sqrt(4.1).
L(4.1)=2+41(0.1)=2.025
Will a function with with f′′(x)<0 be an overestimate or underestimate?
What is an overestimate?
f′(a); slope of the tangent line at a single point.
What is Instantaneous Rate of Change?
Using a table of values: f(1)=2, f(2)=5, f(3)=10, estimate the average rate from x=1 to x=3
What is 4?
A ball’s height is given by h(t)=−16t2+64t h(t) = -16t^2 + 64th(t)=−16t2+64t. What is the IROC at t=2t = 2t=2?
What is 0?
Find the line tangent to f(x)=ln(x) at x=1
y - 0 = 1/x(x-1)
Given f(x)=sqrt(x), is the linear approximation at x=4 of sqrt(4.1) an overestimate or underestimate?
What is an underestimate?
Based on concavity: concave up → under, concave down → over.
What is Over/underestimation?
Estimate the AROC from a table: f(2)= 5.4, f(5)=8.1
What is 0.9?
f(x)=3e2x, find f′(x). Then evaluate f′(0).
What is f′(x) = 6e2e and f'(0) = 6?
Estimate ln(1.05) using linearization.
L(1.05)=1.05−1 = 0.05
Is the tangent line for f(x)=ex at x=0 an over or underestimate at x=0.1?
What is an overestimate?
L(x)=f(a)+f′(a)(x−a); approximation of a function near x=ax = ax=a.
What is Linearization?
Given a function increasing but concave down from x=0 to x=3, is AROC less than or greater than IROC at x=1.5?
What is less than?
The population of bacteria grows according to P(t)=500e0.3t
What is the IROC at t=4t = 4t=4?
What is 498?
Use the linearization of f(x)=ex at x = 0 to estimate e0.1
L(0.1)= 1+ 0.1=1.1
Use the tangent line to f(x)=ex at x=0 to approximate f(0.2). Is this an overestimate or underestimate?
What is an underestimate?