General
Anti/Derivatives
Anti/Derivatives
What is the derivative of
x3-4x2+12?
3x2-8x
where does the graph of 3x2-5x-2/x2-7x+10 have a removable discontinuity
at x=2
y=x and y=2-x and y=0. What is the integral to find the area of the intersecting equations
(1,0)∫x-0dx + (1,2)∫2-x-0dx
solve for y. dy/dx=ky
y=Cekt
When is f(x) concave up?
∫1/x=?
ln│x│+c
Find the average rate of change on the given interval
t (minutes) 0 3 4 12 13
s(t) (feet) -2 4 -7 5 10
on [3,13]
0.6 ft/min
find the area bounded by y=2-x2 and y=x
solve for y. dy/dt=k(y-m)
y=Cekt+c+m
If the position of a particle is represented by the equation (t-8)2+t, where "t" is time, what is the velocity at t=7?
-1
If f(x)=6x2(x3-2), what is f'(x)?
30x4-24x
f'(x)=sin(x). On the interval (-5pi/2,5pi/2), at what x-value(s) does x have a relative minimum?
x=-2pi,0,and 2pi
A rectangles length is growing at a rate of 3 in/sec and its width is decreasing at a rate of 2 in/sec. How fast is the area changing at the moment its length is 10 inches and its width is 6 inches.
da/dt=-2in2/sec
People enter a zoo represented by the function E(x). People exit the zoo represented by the function L(x). Time, t, is measured in in hours after opening
Explain the meaning of N(x)=∫ E(x)-L(x) on [0,t]
n(x) means the amount of people in the zoo after t hours from opening.
What is the derivative of arcsin(x)
1/(1-x2)1/2
∫x3-1=?
x4/4-x+C
what is the slope field of dy/dx=2x/y
plot 16 points, 4 per quadrant.
y=x2 and y=x1/2. With Square cross section perpendicular to the y-axis, set up the integral to find the volume of the intersecting boundaries
∫(y1/2-yy)2 on [0,1]
in four months, Sales drop from 100,000 units/month to 80,000 units/month. If its decaying exponentially, what are the sales after 2 months?
about 71,554 units/month
What is the definition of an Inflection point?
Points of inflection on f(x) are relative extrema on the graph of f'(x)
∫3x2(x3+2)4dx=?
(x3+2)5/5 + C
x.) 1 2 3 6 10
f.) 6 9 10 -1 3
g.) 2 3 4 6 11
f and g are both continuous for all real numbers, and g is strictly increasing. The function h is given by h(x)=g(f(x)). Explain why there must be a value r for 1<r<3 such that h(r)=8.
since f and g are both continuous, that means h is also continuous. And since h(1)<8<h(3), there exists a value "r" such that h(r)=8 on [1,3]
A rectangle has a width of 16 inches, and a length of 18 inches, What is the optimum volume of the box?
361.097 inches3
After an exam, students begin posting memes at a rate of m(t)=231sin(pi(t)/24) for 0<t<24, where m is measured in students per hour and t in hours after the exam ends. after 12 hours, the teacher begins cancelling the grades of those who have made memes at a rate of C(t)=3e0.26439t^2/24 for c is is measured in students per hour and t is in hours. Let S(t) equal those who have posted memes, but have not yet had their scored canceled. At t=19, is S(t) getting larger or smaller, and explain?
S(19) is getting smaller because S'(19)=-19.4296, which is less than 0.
Oil is leaking out of a pipe, and the whole is progressively getting bigger
t (time) 1 2 3 4 5 6 7Rate (liters/min) 6 10 15 24 20 16 2
estimate the amount of oil spilled from t=1-7 using the trapezoidal rule with 6 equal sub intervals.
89 Liters