Each of the following limits represents a derivative
f '(a).
Find
f(x)
and
a.
lim h→0
| (5 + h)3 − 125 |
| h |
| f(x) | = | |
| a | = | 5 |
lim x→2
| x3 − 8 |
| x − 2 |
| f(x) | = | |
| a | = | 2 |
lim h→0
| 52 + h − 25 |
| h |
| f(x) | = | |
| a | = | 2 |
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2. Determine if the following series converge or diverge. Justify your answers, citing any appropriate tests for convergence that you use.
(a) sigma^infinity_n=1 n + 2/(n^5/3 + n + 5 )
(b) sigma^infinity_n=1 (1 − 1/ n)^n ^2
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1a. Find the equation of a tangent line to the curve f(x) = ln(2−x)/x + 3x at the point (1, 3).
1b. Suppose the following function is defined implicitly by the equation, Find dy/dx x^2 − 3y^2 + 6e^x = 4x^2y + 5
2. Using only the definition of derivative as a limit, calculate f(x) where f(x) = 1/x − 5
3. One thousand dollars is invested at a rate of 3%
a) How much money will be in the account in 8 years if interest is compounded every 3 months?
b) How long will it take for the initial investment to double if interest is 3% and compounds continuously?
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Check all of the following that are true for the series
∑n=1∞(n−3)cos(n*π)n^2
A. This series converges B. This series diverges C. The integral test can be used to determine convergence of this series. D. The comparison test can be used to determine convergence of this series. E. The limit comparison test can be used to determine convergence of this series. F. The ratio test can be used to determine convergence of this series. G. The alternating series test can be used to determine convergence of this series.
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1, Let
f(x)=x3/3−0.5x2−6x
Find the interval where f(x) is decreasing.
Enter your answer for the left endpoint in the first textbox, and
right endpoint in the second textbox:
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Bob throws a ball straight up with an initial speed of 46 feet per second from a height of 77 feet.
(a) Find parametric equations that describe the motion of the ball as a function of time.
(b) How long is the ball in the air?
(c) When is the ball at its maximum height? Determine the maximum height of the ball.
(d) Simulate the motion of the ball by graphing the equations found in part
(a). Assume Bob stands at horizontal position 0, and use gequals=3 ft/sec/sec.
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Sketch the graph of the function ?(?)= (2x2-5x+2)/(x+1)2, given the derivatives:
? ′(?) = (9?−9)/(?+1)3 and ? ′′(?) = (36?−18)/(?+1)4
Your sketch should consider the following:
● x- and y-intercepts, if any.
● Horizontal and vertical asymptotes, if any. (Show the computation
of any relevant
limit.)
● Intervals over which ?(?) is increasing/decreasing.
● Intervals over which ?(?) is concave up/down.
● Relative (local) maximum/minimum points and points of inflection,
if any. Identify
these clearly on the sketch.
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Obtain the general and particular solution of the DE d2 y(x) / d x2 + (1/2) (dy(x) / dx) - (1/2)y(x) = x + 1
By use of the variation of parameters method. Evaluate all coefficients please.
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Two cars start moving from the same point. One travels south at 56 mi/h and the other travels west at 42 mi/h. At what rate is the distance between the cars increasing three hours later?b\
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For the following questions:
A. Draw a picture
B. Compute the different lengths
C. Compare lengths or replace into formula for sphere
D. Explain your final results in complete sentences
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Evaluate the improper integral or state that it is
divergent.
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Parameterize the given curves:
a) The ellipse in the plane x=11 centered at the origin and going thru the points (+/- 7,0) and (0,+/- 3) from the perspective of the yz-plane.
b) The circle in the plane z=-2 with radius 7 centered at the origin
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. Let f(x) = 3x^2 + 5x. Using the limit definition of derivative prove that f '(x) = 6x + 5
Then, Find the tangent line of f(x) at x = 3
Finally, Find the average rate of change between x = −1 and x = 2
In: Math