Limits and Continuity
lim x->3 (x+2)
5
Find the derivative of f(x)=3x3-5x2+2x-7
f'(x)=9x2-10x+2
int (3x2-4x+1)dx
x3-2x2+x+C
f(x)=4x3-9x2-12x-3. Find critical points.
x= -1/2, 2
Does this series diverge or converge?
∞∑n=0 2/(3+5n)
divergent series
lim x-> 0 (sinx/x)
1
y=sinx+ln(5x). Find d2y/dx2.
y''=-sinx - 1/x2
int sec2x dx
tan(x) + C
Does the line tangent to the graph of 𝒉 at the given value of 𝒙 lie above or below the graph of 𝒉? h'(x) = (x2-4)/x at x=2
Below because h''(2)>0.
1/2,3/4,7/8,15/16,31/32
sequence converges or diverges?
converges to 1
Determine whether the function f(x) = (x^2+1)/(x-1) is continuous at x = 1?
Not Continuous at 1
Let 𝑓(𝑥) = 𝑥 ∙ 𝑔(ℎ(𝑥)) where 𝑔(4) = 2, g'(4) = 3, ℎ(3) = 4, and h'(3) = −2. Find 𝑓′(3).
-16
int x3 * cos(x4) dx
(sin(x4))/4 + C
Write an equation of the line tangent to y=x3-3x2-4 at its point of inflection.
y+6=-3(x-1)
Name 3 test to determine if a function diverges or converges.
comparison test, the integral test, the ratio test, limit test, alternating series test
Find the lim x->oo (3x2+5x-2)/(2x2-4)
3/2
If f(x)=x3 and g(x)=sinx, find (f * g)'(pi/2).
3pi^2/4
int12 (x)/(x2+1)2 dx
3/20
Let h be the function given by h(t)=70-15cos(pi*t/3) + 5sin(pi*t/4) for 0<=t<=5. At what value of t is h increasing most rapidly?
1.343
Find the first 3 terms from the series ∞∑n=1 (2−n/n2+1)
1/4+1/20+1/80
Given the piecewise f(x)=
2x+1 if x<=0
x2+1 if x>0
Determine whether f(x) is continuous at x=0. Show work.
f(x) is continuous at x=0
If arctan y=lnx, then dy/dx=
(1+y2)/x
A particle travels with velocity modeled by the function v(t)=t2-8t+12. What is the distance traveled by the particle from t=0 to t=6?
64/3
Find points of inflection. f(x)=2x4-8x+3
No points because it doesn't change signs.
Determine if the following series converges or diverges.
∞∑n=1 (31−2n/n2+1)
1/9<1
Converges