Limits
Evaluate: lim(x→2)(x+3)
5
d/dx(x³)
3x²
What does f′(x) represent?
Instantaneous rate of change
∫2 dx
2x+C
Derivative of a constant?
0
lim(x→0)(sin x)/x
1
d/dx(sin x)
cos(x)
Critical points occur when...?
f′(x)=0 or undefined
∫x² dx
x³/3+C
Antiderivative of 1/x?
ln|x|+C
lim(x→∞)(3x²+1)/(x²−4)
3
d/dx(ln x)
1/x
f′ changes + to −. What occurs?
Local maximum
∫cos(x) dx
sin(x)+C
Theorem connecting derivatives and integrals?
Fundamental Theorem of Calculus
Does lim(x→0) 1/x exist?
No
d/dx[(x²+1)(x−2)]
3x²−4x+1
f″ > 0. Concave up or down?
Concave up
∫₀² x dx
2
Slope of tangent to y=x² at x=3?
6
Vertical asymptote of f(x)=1/(x−4)
x=4
d/dx[(x²+1)/(x−1)]
(x²−2x−1)/(x−1)²
Max area with fixed perimeter: optimization?
Optimization problem
Meaning of a definite integral?
Net area/accumulation
If velocity is positive and increasing, what is acceleration?
Positive acceleration