derivatives
x^2
2x
What are the 5 ways to evaluate limit?
1) direct substitution
2) Rationalizing
3) factoring
4) Using trig
5) value of table
6) L'Hôpital's Rule
If a function f(x) is continuous at the point x = 3 and f(3) = 7, what is the value of the limit lim(x -> 3) f(x)?
also 7
What is the vertical asymptote of the function f(x) = 1/(x - 2)?
vertical asymptote of the function is x = 2.
s(t)=3t^3−4t^2+2t−1 meters at time t in seconds. Find the instantaneous rate of change of the position with respect to time at t=2.
s′(t)=9t2−8t+2
s′(2):22 m/s
2x^5+5x^3
10x^4+15x^2
What are some common types of discontinuities in functions?
discontinuities include removable, jump, and infinite discontinuities.
Consider a function g(x) defined over the interval [1, 5]. If g(1) = 2 and g(5) = 8, and g(x) is continuous over the entire interval, what is the average rate of change of the function over this interval?
(g(5) - g(1))/(5 - 1) = (8 - 2)/4 = 6/4 = 3/2 or 1.5.
Determine the limit as x approaches 3 for the function g(x) = (x^2 - 9)/(x - 3).
The limit as x approaches 3 for the function g(x) = (x^2 - 9)/(x - 3) is 6.
v(t)=5t. Find the instantaneous rate of change of velocity at t=4 seconds.
take the derivative so 5
(x^2+3x)/(x-1)
(x^2-x-3)/(x-1)^2
What does the Squeeze Theorem state, and how can it be used to determine limits? (traditional way)
The Squeeze Theorem states that if two functions are on top of each other, and the two bounding functions have the same limit as the target function, then the target function also has that limit.
To remove the discontinuity at x = 2 for the function g(x) = (x^2 - 4)/(x - 2), we should assign the value g(2) = 4.
For the function h(x) = (3x^2 + 2x - 1)/(2x^2 + x - 5), find the horizontal asymptote as x approaches infinity.
The horizontal asymptote of the function h(x) = (3x^2 + 2x - 1)/(2x^2 + x - 5) as x approaches infinity is y = 3/2.
A bicycle travels 15 miles in 1.5 hours. Find the average speed of the bicycle.
total distance / total time = 10
sin(2x^3+1)
6x^2cos(2x^3+1)
Solve Limit as x approaches 0 of (3x)/(x)
3
What are the three ways checking for continuity?
1) f(c) exists
2) limit as x approaches c of f(x) exists
3) if number 1 equal to number 2
Calculate the average rate of change of the function h(t) = 2t^2 - 3t + 1 over the interval [1, 2].
The average rate of change of h(t) = 2t^2 - 3t + 1 over the interval [1, 2] is 3.
bicycle A travels 15 miles in 1.5 hours. Bike B travels 20 miles in 3 hours. Find the average speed of the bicycle.
bike a = 10
Bike b= 6.67
(x^2+1)(3x-2)
Calculate the limit as x approaches 2 of the function of (x^2-4)/(x-2)
4
To confirm continuity over an interval, what conditions must a function satisfy, and how can we check for continuity on a closed interval?
Calculate the limit as x approaches negative infinity for the function k(x) = (4x^3 + 2x^2 - 3)/(3x^3 - 5x + 1).
4/3.
A train travels a distance of 300 miles. It takes 4.5 hours to complete the journey. However, during the first 2 hours, it travels at a constant speed of 60 miles per hour, and during the remaining time, it travels at a constant speed of 80 miles per hour. Find the average speed of the train for the entire journey.
1) convert units
For the first 2 hours at 60 mph, the distance covered is 60 mph×2 hours=120 miles60mph×2hours=120miles.
For the remaining 2.5 hours at 80 mph, the distance covered is 80 mph×2.5 hours=200 miles80mph×2.5hours=200miles.
and then calculate time and distance
320 miles / 4.5 hours = 71.11 miles per hour