Limits and Continuity
Find the limit of x2-5x-50/x-10 as x approaches 10
15
Find dy/dx of y=x3/3+x2/2+x
y'=x2+x+1
When looking for absolute extrema, what two types of points are possible candidates?
1. endpoints
2. critical points
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A person 2m tall walks toward a lamp post on level ground at a rate of 0.5m/sec. The lamp on the post is 5m high. How fast is the length of the person's shadow decreasing when the person is 3m from the post?
The shadow is decreasing at a rate of 1/3m/sec.
Evaluate the integral from 0 to 1 of (x2+ -sqrtx)dx
-1/3
Find the limit of 1-cosx/x as x approaches 0.
0
Find the equation of the line perpendicular to the tangent to the curve y=x3-3x+1 at the point (2,3)
y-3=-1/9(x-2)
Find the extrema on the interval and where they occur given the following equation: g(x)=ln(x+1) over the interval [0,3]
-Absolute max of ln4 at x=3
-Absolute min of 0 at x=0
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A boat is being pulled into a dock by attaching to it and passing through a pulley on the dock, positioned 6 meters higher than the boat. If the rope is being pulled in at a rate of 3meters/sec, how fast is the boat approaching the dock when it is 8 meters from the dock?
The boat is being pulled in at a rate of 30/8meters/sec.
Evaluate the integral from 1/2 to 1 of 1/sqrt(1-x2)dx
pi/3
Name the three types of discontinuities:
1. Removable (hole)
2. Discontinuity due to vertical asymptote
3. Jump discontinuity
Find dy/dx of y=sin5x-sec(10x)+3tan(x/8)
y'=5sin4xcosx-10sec(10x)tan(10x)+3/8sec2(x/8).
Find the x-values of all relative extrema and the intervals on which the function is increasing and decreasing for the following equation: y=2/x
-no relative extrema
-never increasing
-decreasing over (negative infinity, 0) U (0, infinity)
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A ladder 10 meters long is leaning against a vertical wall with its other end on the ground. The top end of the ladder is sliding down the wall. When the top end is 6 meters from the ground, it is sliding down at 2m/sec. How fast is the bottom moving away from the wall at this instant?
The bottom of the ladder is moving away from the wall at a rate of 3/2meters/sec.
Evaluate the integral of (ex/1+2ex)dx using u-substitution.
1/2ln|1+2ex|+c
Find the limit of the square root of x+19 minus the square root of 19 over x as x approaches 0.
1/2 square root of 19
Find the equation of the line tangent to the graph of y=sin(x)+3 at x=pi
y-3=-1(x-pi)
Suppose you are given a formula for a function f. How do you locate inflection points? (explain in words)
Take the second derivative and set it equal to 0 and undefined. Use a sign chart to look for places where the second derivative changes sign.
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A man 6ft tall walks at a rate of 5ft/sec toward a streetlight that is 16ft above the ground.
a) At what rate is the tip of his shadow moving?
b) At what rate is the length of his shadow changing when he is 10 feet from the base of the light?
a) The tip of the shadow is decreasing at -8ft/sec.
b) The length of the shadow is decreasing by 3ft/sec.
Evaluate the integral 1/x2cos(1/x)dx using u-substitution.
-sin(1/x)+c
Find the limit of sin of x+3(pi)x2/2x2 as x approaches infinity
-1
Find the rate of change of the function s(t)= the square root of t2+2t+8 at the point (2,4)
3/4
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Find two positive numbers such that their product is 192 and the sum of the first plus three times the second is a minimum.
The numbers are 8 and 24.
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An airplane is flying towards an airport at a constant height of 6km above the ground. If the distance s between the airplane and the airport is decreasing at a rate of 400km per hour when s=10km, what is the horizontal speed of the plane?
The horizontal speed of the airplane toward the airport is 500km/hr.
Evaluate the integral 1/(x)sqrt(4x2-36)dx using u-substitution
1/6sec-1(1/3x)+c