Limits
Evaluate:
limx→3(2x+5)
11
f(x) = x2
2x
What does a positive derivative mean about a function?
The function is increasing.
∫ x dx
(x2 / x) + c
dy/dx = 3
y = 3x + c
limx→2 (x2 - 4)
0
f(x) = 3x3 - 2x + 7
9x2 - 2
What does a negative second derivative indicate?
The graph is concave down.
∫ 4x3 dx
x4 + c
What is a slope field used for?
To visualize solutions to differential equations.
limx→1 ((x2 - 1) / (x - 1)
2
f(x) = sin x
cos x
If velocity is increasing, what can be said about acceleration?
Acceleration is positive.
∫ cos x dx
sin x + c
dy/dx = 2x
y = x2 + c
limx→2 ((sin x) / x)
1
f(x) = ex
ex
Find the critical points of x2 - 4x
x = 2
∫02 x2 dx
8/3
What does "separable" mean in differential equations?
Variables can be separated onto opposite sides of the equation.
limx→∞ ((3x2 + 1) / (x2 - 5))
3
f(x) = x2 ln x
2x ln x + x
A particle’s position is:
s(t) = t3 - 6t2 + 9t
Find the acceleration.
a(t) = 6t - 12
Find the area under the line y = x from x = 0 to x = 3.
9/2
dy/dx = y
y = cex