DERIVATIVES
FIND THE DERIVATIVE
f(x) = 4x3+10x4
f'(x) = 12x2+40x3
Find the average value of the function y= 3x2+4 on the interval [-1,2]
7
Radioactive Cricellium decays with the equation
A = A0e-0.7t where t is measured in years and A is measured in grams.
Find the half life of Cricellium.
.990 years
An equation of the line tangent to y= x3+3x2+2 at its point of inflection is...
y= -3x+1
In the definition of Riemann Sum, what does "n" represent (in terms of rectangles)
The number of rectangles
FIND THE DERIVATIVE
f(x) = -sin2(4x+1)
f'(x) = 8sin(4x+1)cos2(4x+1)
Evaluate the integral ex+2 dx from [3,7]
7954.671
Let k(x)= x2-5x+4
Find the AROC of k over the interval [3,5]
3
Given f''(x)= 3x2(x-4)5(x+3), what are the x coordinates of points of inflections for the graph of f(x)
x = -3
x = 4
Differentiability implies...
Continuity
FIND THE DERIVATIVE
f(x) = esin(x) + ln(x)
f'(x) = cos(x)esin(x)+(1/x)
The region bounded by y= 3x-x2 and the x-axis is revolved about the line y= -2
What is the volume of the solid?
81.996
Find all x-values at which f(x)= (5x-1)/(25x2-4) is discontinuous
x= 2/5
Find the absolute extrema of f(x)= 2x3-3x2-12x+5 on the interval -2<x<3
*can use calculator*
Max of 12 at x= -1
Min of -15 at x= 2
When is a function concave down?
When f'(x) is decreasing
OR
When f''(x)<0
FIND THE ANTI-DERIVATIVE
f'(x) = 1/(x2+1)
f(x) = sin-1 + c
Determine the area of the region bounded by
g(x) = 2x - sin(x) and the x-axis on the interval π<x<2π
3π2+2
Find the absolute extrema of f(x)= x3-12x+23 on the interval -5<x<3
Absolute min of -42 at x= -5
Absolute max of 39 at x= -2
Find the perimeter of the region in the first quadrant bounded by y= 16-x2, the x-axis, and the y-axis
*may use calculator*
36.819
If a function is continuous on the interval [a,b] and differentiable on the same interval, what is gauranteed by the mean value theorom?
A number 'c' is gauranteed in the interval [a,b] such that f'(c)= (f(b)-f(a))/(b-a)
FIND THE H'(1)
g(x) = 3x2-2x
f(x) = x3
H(x) = g(f(x)) + g(x)
H'(x) = 16
A particle moves along a line so that its position at any time is given by the function x(t)= tan(t) where x(t) is measured in feet and t is measured in seconds.
Find the instantaneous velocity when t = 2π/3
4 ft/sec
A rocket blasts off with a constant acceleration of 40 ft/sec2. After 2 seconds the rocket is 100 ft. high. Use antiderivatives to find an equation for the height of the rocket. (hint, the rocket is not moving at t=0 seconds)
h= -20t2+180
Consider f(y)= y2-8 and g(y)= 2y
Find the 2 points of intersection of the graphs f and g
(8,4) and (-4,-2)
Define the second part of the fundamental theorom of calculus
If a function f is continuous on [a,b] and F is any anti-derivative of f, then
b
S f(x) dx = F(b) - F(a)
a