True or False
A function has to be continuous at
x=a
if the
lim_(x->a) f(x)
exists.
What is, not necessarily. A function can have a removable discontinuity for the limit to exist.
Evaluate the limit algebraically or explain why the limit does not exist:
lim_(x->0)(3x^2-2x+4)
What is 4?
Find the derivative of f(x)=(4-x^2)^3
What is
-6x(4-x^2)^2
The critical points and the local and absolute extrema of f(x)=3x^4-4x^3=12x^2+6 on [-3,3] .
What is
critical points: x=-1, x=0, x=2;
local minima points: (-1, 11) and (2, -26)
local maximum: (0, 6)
absolute maximum: (-3, 249)
absolute minimum: (2, -26)?
Is F(x)=x^2+4x+1 the antiderivative of
f(x)=2x+4
What is Yes?
If there is a vertical asymptote at x=a for the function y=f(x) then f(x) is undefined at the point x=a
What is True? The point is not in the domain of the function.
Evaluate the limit algebraically or explain why the limit does not exist:
lim_(x->pi/2)cotx/cosx
What is 1?
Find the derivative of
x^2y=y+2+xysinx
What is
dy/dx=(y*(sinx+cosx))/(2x-1-sinx)
?
The intervals where y=x+sin2x is increasing, decreasing, concave up, and concave down on [0,pi]
What is increasing on (0,pi/3)U((2pi)/3, pi)
decreasing on (pi/3, (2pi)/3)
concave up on (0,pi/2)
concave down on (pi/2,pi)
The antiderivative of f(x)=e^x-3x^2+sinx
What is F(x)=e^x-x^3-cosx+C ?
A continuous function has a continuous derivative.
What is False? A function itself might be continuous, but the derivative might not exist at some points.
Evaluate the limit algebraically or explain why the limit does not exist:
lim_(x->2) (3x^2-2x-8)/(x^2-4)
What is 4?
The third derivative of
y=(3x+2)^2
What is 0?
The derivative of f(x)=3^sin(2x)
What is dy/dx=2cos(2x)*ln3*3^sin(2x)
The derivative
d/dx int_1^(x^3)sqrt(4-t^2)dt
What is
3x^2*sqrt(4-x^6)
If f'(c)=0 there is a maximum or minimum at x=c
What is False? If the derivative does not change sign at the critical point, that point is neither maximum nor minimum.
Evaluate the limit:
lim_(x->0)(3^x-2^x)/x
What is ln(3/2) ?
The equation of the tangent line to the y=arccosx+x at x=0
What is
y=-x+pi/2
?
The derivative of
y=(x+11)/(x^2-4)^(1/3)
(use logarithmic differentiation)
What is
dy/dx=1/(x^2-4)^(1/3)-(2x(x+11))/(x^2-4)^(4/3)
You are merging onto the highway, accelerating at a constant rate of
12 (ft)/sec^2
. How long does it take you to reach merging speed at 60 mph? How far doe the car travel to reach the merging speed? (1 mi=5280 ft)
What is v(t)=12t , t=22/3, d=322.67 ft
All continuous functions have an antiderivative.
What is True? Continuous functions are integrable.
Evaluate
lim_(x->0^+)x^(1/cosx)
What is 0 ?
The differential of y=x^2-5x-6 evaluated at x=2 and dx=0.1
What is
dy=(2x-5)dx and dy=-0.1
The side of a cube increases at a rate of 0.5 m/sec. Find the rate at which the volume of a cube increases when the side of the cube is 4 m.
What is 24 m^3/sec
Identify the roots of the integrand to remove absolute values, then evaluate using the Fundamental Theorem of Calculus:
int_0^pi abs(cost)dt
What is int_0^((pi)/2) cosx \ dx+int_((pi)/2)^(pi)(- cosx )\ dx
=2?