Limits
lim x->1 (x²-1)/(x-1)
What is 2
A function must be __________ at a point before it can be differentiable there.
Continuous
Given the function f(x)= (2x² + 9x - 5) / [(x +5) (x - 1)] find the discontinuity and label as a removable, jump, or infintie discontinuity.
What is removable at x= -5, and infinite at x= 1
Using differentials, find the approximate value of
32.151/5
2.0019
lim_(x->9)f(x)=7 and lim_(x->9)g(x)=2 Solve lim_(x->9){f(x)+g(x)}
What 9
If f(x)=∣x∣, then f(x) is __________ at x=0.
Non differentiable
Given the function, f(x)=(-5x-15)/(2x^2-18 , find all holes and vertical asymptotes. What kind of discontinuities would these be?
What is a hole at x=-3 (removable), and a vertical asymptote at x=3 (infinite)
Using differentials, find the approximate value of (35.8)1/2
5.9833
lim_(t -> -3) (t² - 9) / (2t² + 7t +3)
What is
6/5
If the left-hand derivative is −3 and the right-hand derivative is 5, then the function is
not differentiable at the point
Given the function, f(x) = {(x²-9, x≠3), (7, x=3) :} find any points of discontinuity and label as a removable, jump, infinite, or oscillating discontinuity.
What is removable at x=3
A ladder 13 m long is leaning against a vertical wall. The foot of the ladder is pulled away from the wall at the rate of 2 m/s. Find the rate at which the top of the ladder slides down when the foot is 5 m from the wall.
-5/6 m/s
lim_(x->0)sin(5x)/(8x)
What is
5/8
Determine whether the function
f(x)=∣x−3|
is differentiable at x=3.
not differentiable
Using limits, define k so that f(x) = {(x²-16, x≠3), (k, x=3) :} is continuous at x=3
f(3)=-7
lim_(x->3)f(x)=lim_(x->3)(x^2-16)=3^2-16=-7
The radius of a circular oil spill is increasing at the rate of 4 cm/min. Find the rate at which the area is increasing when the radius is 10 cm.
80π cm2/min
lim x→2 x3−8/x2−4
3
Find the value of aaa such that
f(x)={ax2+1,x<1
3x−1,x≥1
is differentiable at x=1.
a = 1
This value assigned to b will make the function g(x)={(x^3, x<1/2),(bx^2, x>=1/2):} continuous as x=1/2
What is
b=1/2
Water is poured into an inverted conical tank of height 12 m and base radius 6 m at the rate of 9πm3/min. Find the rate at which the water level is rising when the depth of water is 4 m.
9/4