Unit One- Limits/Continuity
limx→2 x ^2 − 4 /x ^2 + x −6
4/5
f(x)= x^8+12x^5-4x^4+10x^3-6x+5
f'(x)=8x^7+06x^5-16x^3+30x^2-6
Find two numbers whose difference is 100 and whose product is a minimum.
X=50 Y=-50
Estimate the area under the graph f(x)=x^2 on the interval 0 to five taking the same points as left endpoints using five subintervals.
Area=30 Units^2
Find the area between the curves y^2=-x-2, y=x+1, y=-2, y=2
Area=28/3
lim h→0 (1 + h)^2 − 1 /h .
2
With the Motion Function, find the acceleration
s(t)=4t+.8t^2+.333t^3
a(t)=1.998t+1.6 m/s^2
Find the antiderivative of f(x)=x^3
F(x)= 1/4x^4
∫( x^2+5)^3 2xdx
=1/4(x^2+5)^4 +c
Let R be the region enclosed by the graphs of f(x) = 16 − (x − 2)^2
, g(x) = 4 − 2x, the y-axis, and the line x = 3 as shown in the diagram below. Find the area of R.
42
limx→1 x ^2 − x /x ^2 + 2x − 3
1/4
Differentiate f(x)=(x^4+9)^7
f'(x)= 28x^3(x^4+9)^6
A rectangle with horizontal and vertical sides has one vertex at the origin, one on the positive x-
axis, one on the positive y-axis, and one on the line 2x+y=100. What is the maximum area of the rectangle?
1250 Units^2
∫ 4x-10/(x^2-5x+1)^3 dx
=-1/(x^2-5x+1)^2 +c
The region R in the first quadrant enclosed by the graphs of y = 2x and y = x^2 is the base
of a solid. For this solid, the cross sections perpendicular to the y-axis are squares. Find the volume of
the solid.
Volume= .533
The graph of y=x^2-9/3x-9 has
A removable discontinuity at X=3
find dy/dx if xy-4=0
-y/x
Find two positive numbers whose sum is 20 and whose product is as large as possible.
Y=10 X=10
Given the equation f(x)=-1/2x^2+6 and the interval [-1,3] estimate the area under the curve using LRAM4
Area= 21
Let R be the region in the first quadrant bounded by f(x) = e^x + 3 and g(x) = ln(x + 1) from
x = 0 to x = 2. What is the area of R?
11.093
For what value(s) of a is f(x) = {x^2, x ≤ 1
ax + 2, 1 < x ≤ 3
continuous at x = 1?
a=-1
if A=x^2 and dx/dt=3 Find dA/dt when x=10
dA/dt=60
Suppose you had 102 m of fencing to make two side-by-side exclosures as shown. What is the maximum area that you could enclose?
433.5 m^2
With the Velocity function given for a particle moving in a line find the displacement.
v(t)= 3t-5 x<t<3
-3/2
Let f and g be the functions given by f(x) =1/4+ sin(πx) and g(x) = 4^−x
. Let R be the shaded region in the first quadrant enclosed by the y-axis and the graphs of f and g, and let S be the shaded region in the first quadrant enclosed by the graphs of f and g, as shown in the figure below.
(a) Find the area of R.
.065