Limits
What is the limit of f(x)=3x+2 as x approaches 1?
5
The derivative of 3.
What is 0?
The derivative of y?
What is dy/dx?
What does it mean for a function to be increasing or decreasing based on its derivative?
A function is increasing if it's derivative is positive and decreasing is its derivative is negative
What is the purpose of the Mean Value Theorem?
there is a point where the instantaneous slope of the tangent = the avarage rate of change over the interval
How do you evaluate lim x→0 sinx/x?
1
What is the derivative of f(x)=x3?
3x2
Differentiate the equation x2+y2=25 implicitly.
-x/y
How can you use derivatives to find the maximum or minimum of a function?
Locate points where f'(x) = 0 or is undefined then use the second derivative test to confirm
State the Fundamental Theorem of Calculus.
the value of any function is the rate of change (the derivative) of its integral from a fixed starting point up to any chosen end point.
If g(x)<f(x)<h(x) on (a,b) and limx->0g(x)=1 and limx->0h(x)=1 then the limx->0f(x) must be?
What is 1?
Find the derivative of f(x)=sinx
cosx
If x3+y3=6xy, find dy/dx.
dy/dx= 3x2-6y/6x-3y2
Describe the relationship between the first derivative and the slope of a tangent line.
the first derivative of the funtion at a point gives the slope of the tangent line to the curve at that point
How do you determine concavity using the second derivative test?
The second derivative test determines concavity. If f"(x) is less than 0 the function is concave up and if f"(x) is less than 0 than the function is negative
Find limx→2(x2−4)/(x−2)
4
Use the product rule to differentiate f(x)=x2⋅lnx
f(x)=2xln(x)+x
Differentiate the equation sin(xy)=x+y implicitly.
cos(xy)(y+x(dy/dx))=1+(dy/dx)
How can you interpret the derivative of a position function?
f' of a position function represents velocity which indicates speed and direction of motion
limx->0(1-cos(2x)/(5x)
What is 0?
Explain the concept of one-sided limits and calculate limx→3 −(x2−9)/(x−3)
6
What is the second derivative of f(x)=ex+3?
ex
Solve for dy/dx in the equation x2y+3y=7.
dy/dx = -(2xy/x2+3)
Explain how derivatives are used in related rates problems with an example.
in a related rates problem, derivatives represent the rate of change of quantities with respect to time. EX: A=(pi)r2 with respect to time relate dA/dt to dr/dt
The vertex of x2+2x+1
What is (-1,0)?