Limits
What is the lim as x approaches 3 of (2x^2 - 5x + 1)?
4
Find the derivative of 4x^5- 3x^2+7x-9
Which theorem states that if a function f is continuous on a closed interval [a, b], then for every value x between f(a) and f(b), there exists at least one x in [a, b] such that f(c) = x ?
Intermediate Value Theorem (IVT)
What are two methods used to find area and volumes?
Disk and Washer Methods
The velocity vector v(t) of a particle moving in the plane whose position vector is given by p(t) = <t^3, e^2t>
<3t^2,2e^2t>
What is the limit as x approaches 4 of (x^2- 16) /(x-4)?
8
Find the derivative of f(x)= 4x^3-5ln(x)+e^x
12x^2-5/x+e^x
This calculus theorem guarantees that if a function is continuous on a closed interval [a, b], it must attain an maximum and a minimum value on that interval.
Extreme Value Theorem
The definite integral that represents the area of the region bounded by y = x^2, the x-axis, x = 0, and x = 3.
9
What is the formula for speed?
sqrt( dx/dt)^2+ (dy/dt)^2
What is the limit as x approaches 0 of sin(5x) / 3x?
5/3
Find the 2nd derivative of 3sin(2x)-cos(5x)
-12sin(2x)+25cos(5x)
Explain when there is a local min or max and when there is a point of inflection.
Min or max: change in slope
POI: Change in concavity
What is the volume using the disk method to find the volume of the solid generated by revolving the region bounded by y = sqrt(x), y = 0, and x = 4 around the x-axis.
8pie or ~25.133
The slope of the line tangent to the path of a vector-defined particle at t = 1 if x(t) = t^2 + 3 and y(t) = t^3 - t.
1
What is the lim as x approaches 0 of sin(1/x)
DNE
Find the derivative of arc tan(3x)
3/1+9x^2
The value of c that satisfies the Mean Value Theorem for f(x) = x^2 on the interval [1, 3]?
2
The integral setup used to find the volume of the solid whose base is the region bounded by y = x and y = x^2, and whose cross sections perpendicular to the x-axis are squares.
integral from 0 to 1 of (x-x^2)^2 dx
A particle's position is given by r(t)= (t^3, e^2t). Find the acceleration vector at t=1.
(6,4e^2)
What is the value of the limit as x approaches 0 from the right of (1+2x)^1/x
e^2
Find the second derivative of sin(x^2+1)
F''(x)=-4x^2sin(x^2+1)+2cos(x^2+1)
Using the fundamental Theorom of Calculus find d/dx of the integral from 0 to x^2 of sin(t)dt.
2xsin(x^2)
The integral setup using the washer method to find the volume of the solid formed by revolving the region bounded by y = x and y = x^2 around the horizontal line y = 2.
pie integral from 0 to 1 of (2-x^2)^2 - (2-x)^2) dx
A particle is moving in the xy-plane such that the posistion vector is r(t)= <1/3t^3-3t, e^2t-4>. At t=2 is the speed of the particle increasing or decreasing? Show work.
Increasing