Limits & Continuity
What is the lim k as x approaches k?
K. The limit is approaching K
d/dx (3x^2−x+3)?
6x-1
Calculate the integral of: ∫(-3,0) (2x-5)dx
-24
What is a graphical representation of the solutions of a first-order differential equation called?
Slope Field
n=0 to infinity ∑ ( ⅓)^n converge or diverge?
Converges because the common ratio r=⅓ satisfies r < 1
What are the vertical and horizontal asymptotes (if any) of f(x)= 4/(x^2-1)?
The x axis acts as a horizontal aymptote and x=1 and x=-1 are the vertical asymptotes
d/dx sin(4x^2)?
8xcos(4x^2)
Evaluate the integral using FTC: ∫(1,4) (1/tˆ2)dt
3/4
Is y' = x - xy separable?
Yes
What is the sum of the geometric series n=0 to infinity ∑ 5(½)^n?
10, The sum is a/1-r which is 5/1-½
Is the function f(x) = x² for x ≠ 2,f(x) = 1 for x = 2 continuous at x = 2? Explain why or why not.
No, this is a removable discontinuity at x=2. Check the three conditions for continuity
Differentiate y=x^3lnx
y′ =x²(1+3lnx)
Water is being pumped into a reservoir at a rate of R(t)=120+4t litres per minute, where 0≤t≤50. Find the total amount of water pumped into the reservoir during the first 15 minutes.
2250 liters. Take the integral and evaluate from 0 to 15.
Use Euler’s Method with a step size of 0.2 to estimate y(1.2), where y(x) is the solution to the initial value problem y’=y-2x, with y(1)=0
y=-0.4
Does the series n=1 to infinity 1/n^2 converge or diverge? What test justifies your answer?
It converges by the p-series test since p=2>1
For what x values is the function (x^2+3x+5)/(x^2+3x-4) continuous?
The function is continuous on all values except x=1 and x=-4
Find the derivative of f(x)=cos(5/x)
(5/x^2) sin (5/x)
Evaluate the indefinite integral: ∫(4x+3)ˆ4dx (hint U sub)
1/20(4x+3)ˆ5+C
What is the general solution to y ’ = 2y + 1 (Hint Separable Eq)?
y = Ce^2x − 1/2
Does n=1 to infinity ∑ (n!/5^n) converge or diverge?
Diverges
What is the limit of sin x/(x^2+3x) as x approaches 0?
1/3, First pull out an x from the bottom half of the equation. Then separate the equation into sin x/(x) times 1/(x+3)
Differentiate y=√(13x²-5x+8)
y′ =26x-5/ 2√13x²-5x+8
Use a left Riemann Sum with 5 subintervals to approximate the area under the curve for the function f(x)=1/x on the interval [1,2]. What is the value of L(5)?
L(5)= 0.745635
The curve plotted through a slope field given an initial condition (in the form of a point) is called what?
Solution curve
FInd the interval of convergence for the power series n=1 to infinity ∑ x^n/n?
(-1,1)