Derivatives
f(x) = 3x2-14x3-sin(x2)
What is f'(x)?
6x-42x2-2xcos(x2)
∫(6x5−18x2+7)dx
Integrate
x6−6x3+7x+C
True
A particle's position is defined by x(t)=x4-3x3+2x2-x+7
What direction is the particle moving at time t=4?
The particle is moving to the right
What is the formula for the chain rule?
f'(g(x))g'(x)
f'(x)=sec(x)
Find f"(x)f"(x)=sec(x)tan(x)
∫sin(x)dx
Solve
-cos(x)+C
What are the components of the definition of continuity?
The limit as x approaches c exists and equals f(c)
A particle's position is defined by x(t)=x4-3x3+2x2-x+7
Is the particle slowing down or speeding up at time t=1
The particle is speeding up
f(x)=x2-4x
Find the relative extrema and justify your answer.The graph of f has a relative minimum at x=2 since f'(x) changes from negative to positive at x=2.
Find the derivative of the following:
∫(2x-x2)dx
2x-x2
∫(1/1+x2)
Solve
arctan(x)+C
limx→2(x2)
4
A particle's position is defined by x(t)=x4-3x3+2x2-x+7
Is the particle moving toward or away from the origin at time t=2
Away from the origin
f"(x)=4x3
Is f concave up or down at x=3
Concave up
g(x)=10tan(x)−2cot(x)
Find the derivativeg′(x)=10sec2(x)+2csc2(x)
∫1/(x+5)1/2dx
2(x+5)1/2+C
limx→−5 (x2−25/x2+2x−15)
Solve
5/4
19/3
x2+y3=4
Find dy/dx
-2x/3y2
What rules of calculus are used to solve this problem?
f(x)=6xln(x2) find f'(x)
Chain rule and product rule
Find the average value for
f(x)=8x−3+5e2-x on [0,2]
0.5(5+5e2)
Use L'Hospital's Rule
limx→2(x3−7x2+10x/x2+x−6)
-6/5
What is the parametric formula for arc length?
∫((dx/dt)2+(dy/dt)2)1/2
What components have to be correct for mean value theorem to apply?
For a function f that's differentiable over an interval from a to b to, that there exists a number c on that interval such that f'(c) is equal to the function's average rate of change over the interval.