微積分
極限、導數、最優化與函數性質。
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think about the process the graph shows the speed of your neighbors car…
For the first question: C. Identify the lines that have a slope of zero. For the second question: 3
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(3) f(x) = 2x² + x + 2; 0, 1
If we consider the average rate of change, it is \( 3 \). If we consider the derivative analysis, the function is increasing on \([0, 1]\) with derivative \( f'(x)=4x + 1 \). (Dep…
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category: homework current score: 23.8/43 points score: 0 of 1 point gi…
\(-12\)
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score: 0 of 1 point identify the intervals where the graph is increasin…
The function is increasing on the intervals \((-\infty,-3)\cup(-1,\infty)\) and decreasing on the interval \((-3,-1)\).
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score: 0 of 1 point use the graph of f to evaluate the following: the a…
\(-4\)
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category: homework current score: 19.8/43 points current learning objec…
\(13\)
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score: 0 of 1 point determine the vertex of the quadratic function $y =…
D. \((5,-4)\)
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a disc - shaped target is launched straight up into the sky. the target…
The tangent of the angle \( \theta \) is given by \( \boldsymbol{\tan\theta=\frac{4.9t^2 - 14.5t + 16}{17}} \) (or if considering absolute value, but since \( h(t) < 16 \) for all…
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use the graph of f to find the value of f(-2). f(-2)=\\square
\( \boxed{3} \) (Note: The actual value depends on the graph's details, but this is a typical answer for such a problem. If the graph shows a different value, adjust accordingly. …
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use the graph of f to find the value of f(0). f(0) = \\square
\( \boxed{0} \) (Note: The answer might vary depending on the actual graph's clear representation. If the graph's y - intercept is different, the answer would change. But based on…
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lisc - shaped target is launched straight up into the sky. the targets …
(The problem is incomplete; the actual question is missing. If assuming to solve \( -4.9t^2 + 14.5t = 16 \), discriminant is negative, so no real \( t \).)
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use the graph of f to find the value of f(-8). f(-8)=
\( \boldsymbol{4} \)
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use the graph of f to find the value of f(-9).
\( \boxed{6} \) (Note: The answer depends on the actual graph. If the graph at \( x = - 9 \) has a different \( y \) - value, the answer should be adjusted. The above is a demonst…
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use the vertical line test to determine if y is a function of x in the …
y is not a function of x
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graph the given functions, f and g, in the same rectangular coordinate …
For \( x=-3 \), \( g(x)=0 \); for \( x=-2 \), \( g(x)=1 \)
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graph the given functions, f and g, in the same rectangular coordinate …
For \( x = 4 \), \( f(x)=2 \); for \( x = 9 \), \( f(x)=3 \)
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graph the given functions, f and g, in the same rectangular coordinate …
For \( x = 0 \), \( f(x) = 0 \); for \( x = 1 \), \( f(x) = 1 \) (filling the table: first empty cell (x=0) is 0, second empty cell (x=1) is 1)
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graph the given functions, f and g, in the same rectangular coordinate …
For \( x = 0 \): \( f(0) = 0 \), \( g(0) = -5 \) For \( x = 1 \): \( f(1) = 1 \), \( g(1) = -4 \) For \( x = 4 \): \( f(4) = 2 \), \( g(4) = -3 \) For \( x = 9 \): \( f(9) = 3 \),…
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graph the given functions, f and g, in the same rectangular coordinate …
The graph of \( g(x)=-2x - 3 \) is the graph of \( f(x)=-2x \) shifted vertically down by 3 units. The two lines are parallel (since they have the same slope \( m = - 2\)).
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evaluate the function $f(x) = \\frac{x}{|x|}$ at the given values of th…
a. \(f(6)=1\) b. \(f(-6)=-1\) c. \(f(r^{2}) = 1\) (for \(r eq0\))
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which equation is represented by the graph below? options: y = e^x y = …
\( y = \ln x \) (the option corresponding to \( y=\ln x \))
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which equation is represented by the graph below? the graph is a coordi…
\(y = e^{x}\) (the option with \(y = e^{x}\))
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multiply. 14.8 × 0.001
0.0148
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the three right triangles below are similar. the acute angles ∠f, ∠j, a…
(a) \( \frac{EF}{DF} = \boldsymbol{0.52} \), \( \frac{IJ}{HJ} = \boldsymbol{0.52} \), \( \frac{QR}{PR} = \boldsymbol{0.52} \) (b) \( \sin 58.8^\circ = \boldsymbol{0.86} \), \( \co…
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what is the true solution to $2 \\ln e^{\\ln 5x} = 2 \\ln 15$? \\(\\cir…
B. $x = 3$
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multiply 1.91 × 0.01
0.0191
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use transformations of the standard cubic function, f(x) = x³, to graph…
To graph \( g(x)=x^3 + 5 \), vertically shift the graph of the standard cubic function \( f(x)=x^3 \) upward by 5 units. The key points of \( f(x)=x^3 \) (e.g., \((0,0)\), \((1,1)…
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a town predicts that its population will double in \\(\\frac{e^{\\ln 10…
5
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solve: 3 ln x = ln 216 x=3 x=6 x=72 done
B. \( x = 6 \) (assuming the options are labeled as A. \( x = 3 \), B. \( x = 6 \), C. \( x = 72 \))
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solve: 4 ln x = 2 ln 25 ○ x = 5 ○ x = 25 × x = 625 retry
\(x = 5\) (the option \(x = 5\))
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use transformations of the absolute value function, f(x) = |x|, to grap…
B. Reflection about the x - axis, C. Vertical stretch by a factor of 2, E. Horizontal shift 1 unit to the left, F. Vertical shift 5 units down
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question 7 (2.5 points) what is an absolute monarchy? a monarchy where …
A. A monarchy where the ruler has complete control and claims a divine or hereditary right to rule ### Question 8
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use long division to rewrite the equation ( g(x) = \frac{6x + 19}{x + 3…
\(g(x)=6+\frac{1}{x + 3}\), and the correct graph is C.
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evaluate the expression. 8 - 4 + 3
7
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expand each expression. $ln (2x)^4$ - $4 ln 2 + 4 ln x$ - $4 ln 2 + ln …
A. \(4\ln 2 + 4\ln x\)
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expand each expression. ln 3x = done
\(\ln(3)+\ln(x)\)
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use transformations of the absolute value function, $f(x)=|x|$, to grap…
A. Horizontal shift 3 units to the left, D. Vertical stretch by a factor of 3
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which expressions are equivalent to \\(\\ln x + 2 \\ln 5 + \\ln 1\\)? c…
\(\ln 25x + \ln 1\), \(\ln 25x\) (corresponding to the third and fourth options: \(\boldsymbol{\ln 25x + \ln 1}\), \(\boldsymbol{\ln 25x}\))
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find the vertical asymptotes, if any, and the values of x corresponding…
B. The vertical asymptote(s) is(are) \( x = 0, x = 6 \). There are no holes.
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use transformations of the absolute value function, $f(x)=|x|$, to grap…
B. Horizontal shift 7 units to the right, E. Reflection about the x - axis
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find the vertical asymptote, if any, and the values of x corresponding …
B. The vertical asymptote(s) is(are) \( x = 0, x = 6 \). There are no holes.
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practice graphing exponential functions in base e. identify the graph o…
The Middle Graph
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use transformations of the absolute value function, f(x)=|x|, to graph …
A. Horizontal shift 3 units to the right
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use transformations of ( f(x) = sqrt{x} ) to graph the following functi…
To graph \( g(x)=2\sqrt{x - 5}+4 \) from \( f(x)=\sqrt{x} \): 1. **Shift 5 units right**: Replace \( x \) with \( x - 5 \) (transforms \( \sqrt{x} \) to \( \sqrt{x - 5} \), shifti…
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use technology to find points and then graph the function $y = 2^x - 2$…
Asymptote: \( y = -2 \) Points (e.g.): \( (0, -1) \), \( (1, 0) \), \( (2, 2) \), \( (3, 6) \) (plot these on the graph).
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use technology to find points and then graph the function $y = 2^x - 2$…
\( y = -2 \)
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question 24 of 24 step 1 of 1 solve the following exponential equation.…
Exact form: \(x=\frac{\ln(1684)}{\ln(8)}-1\) Decimal approximation: \(x\approx2.57\)
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begin by graphing the square root function, f(x) = \\sqrt{x}. then, use…
A
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begin by graphing the square root function, ( f(x) = sqrt{x} ). then, u…
D (the graph corresponding to option D)
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a baseball is hit into the air, and its height h in feet after t second…
(a) \(1\) ft. (b) \(2\) sec. (c) \(65\) ft.
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graph the function using the techniques of shifting, compressing, stret…
A. The graph labeled A (with the starting point around \( x = 6 \) and increasing as \( x \) increases)
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use transformations of ( f(x) = sqrt{x} ) to graph the following functi…
C (assuming C is the graph with the square - root shape starting at (0, - 4) and increasing)
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question 23 - of 24 step 2 of 2 an open - top box is to be constructed …
0, 7
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which statement describes an example of a limit on the power of the hou…
A. The president can veto its bills.
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solve the equation using the quadratic formula. $x^2 + 11x + 9 = 0$ the…
\(\frac{-11 - \sqrt{85}}{2},\frac{-11 + \sqrt{85}}{2}\)
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question 23 of 24 step 1 of 2 an open - top box is to be constructed fr…
\( V(x)=x(22 - 2x)(14 - 2x) \)
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simplify. $e^{ln x} = square$ $e^{ln 5} = square$ $e^{ln 3x} = square$
\( e^{\ln x}=\boldsymbol{x} \) \( e^{\ln 5}=\boldsymbol{5} \) \( e^{\ln 3x}=\boldsymbol{3x} \)
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simplify. $ln e = square$ $ln e^{2x} = square$ $ln 1 = square$ done
$\ln e = \boldsymbol{1}$, $\ln e^{2x}=\boldsymbol{2x}$, $\ln 1=\boldsymbol{0}$
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assignment 11.5: solve systems of nonlinear equations score: 13/17 answ…
\((8,0),(-8,0)\)
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evaluate $y = e^x + 1$ for the following values of $x$. round to the ne…
For \( x = -2 \), \( y \approx \boldsymbol{1.135} \); for \( x = 1 \), \( y \approx \boldsymbol{3.718} \); for \( x = 2 \), \( y \approx \boldsymbol{8.389} \)
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what does \\(\\frac{3}{2} \\div \\frac{4}{5}\\) represent in this situa…
A. lekika's height
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complete each number sentence by choosing >, <, or =. (a) 33 5 × 6 (b) …
(a) 33>5×6 (b) 74+15<96 (c) 49-16=33
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18 pts determine whether the series converges or diverges. state the te…
The series converges (by the p - series test with \(p=\frac{3}{2}>1\)) ### Part (b)
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consider the following rational function. $f(x) = \\frac{x^2 + 10}{x^3 …
\( (0, -\frac{5}{32}) \), \( (1, -\frac{11}{63}) \), \( (2, -\frac{1}{4}) \), \( (5, \frac{35}{61}) \) (Other valid pairs are also acceptable, e.g., using different x - values as …
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practice assignment 3.3 rates of change and behavior of gra score: 10/6…
Increasing on the interval(s): \((-3,1)\) Decreasing on the interval(s): \((-\infty,-3)\cup(1,\infty)\)
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consider the following rational function. step 2 of 3: find equations f…
The horizontal asymptote is \( y = 0 \), and there is no oblique asymptote. So the equation for the horizontal asymptote is \( y = 0 \).
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consider the following rational function. $f(x) = \\frac{x^2 + 10}{x^3 …
\(x = 4\)
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what is the approximate value of x in the equation below? \\log_{5}15 =…
-1.317 (the option corresponding to -1.317)
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a student solved the equation below by graphing.\\(\\log_{6}(x - 1) = \…
The curves do not intersect.
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what is the approximate value of q in the equation below? $q + \\log_{2…
0.585 (corresponding to the option "0.585")
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tenisha solved the equation below by graphing a system of equations.\\(…
B. (1.0, 1.4)
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consider the following: \\(\\sum_{n=1}^{\\infty} \\frac{19}{(2n+1)(2n-1…
\(a_1=\frac{19}{3}\), \(a_2=\frac{19}{15}\), \(a_3=\frac{19}{35}\), \(a_4=\frac{19}{63}\) ### Part (b)
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which is equivalent to \\(\\log_{2} n = 4\\)? \\(\\bigcirc\\) \\(\\log …
\(\log n = 4\log 2\) (the fourth option)
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what is the approximate value of x in the equation below. \\log_{\\frac…
-3.396 (corresponding to the first option: -3.396)
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practice assignment 2.4 complex numbers score: 200/250 answered: 20/25 …
\(-2 + 4i\)
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21 mark for review the area of a rectangular region is increasing at a …
A. 0.39
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20 mark for review the quadratic function g models the depth, in meters…
168
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19 mark for review in the xy-plane, the graph of the equation $y = -x^2…
C. \( -\frac{319}{4} \)
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practice assignment 2.4 complex numbers score: 180/250 answered: 18/25 …
\(-120-22i\)
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graph the function $y = 2^{x - 2} + 1$ using the given table of values …
\( 1 \)
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13 mark for review for x > 0, the function f is defined as follows: f(x…
D. Increasing linear
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question 9 of 24 step 3 of 3 consider the following polynomial function…
1
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the following function is given. $f(x)=x^3 - 2x^2 - 9x + 18$ a. list al…
\( 2 \) ### Part (c)
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the function h is defined by $h(x) = 4x + 28$. the graph of $y = h(x)$ …
A. 21
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practice assignment 2.4 complex numbers score: 140/250 answered: 14/25 …
The real number \(a=-24\), the real number \(b = 0\).
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the following function is given. $f(x)=x^3 - 2x^2 - 9x + 18$ a. list al…
\( \pm 1, \pm 2, \pm 3, \pm 6, \pm 9, \pm 18 \) ### Part (b)
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22 mark for review f(x) = 9(4)^x the function f is defined by the given…
B. \( g(x) = 144(4)^{x} \)
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practice assignment 2.4 complex numbers score: 130/250 answered: 13/25 …
\(-8 + 24i\)
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16 mark for review the function f is defined by f(x) = (x - 6)(x - 2)(x…
76
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for the polynomial function f(x) = -2x⁴ - 6x², answer the parts a throu…
### Part a: The x - intercept(s) at which the graph touches the x - axis and turns around is \( 0 \) (the point \( (0,0) \)) ### Part b: There are no x - intercepts at which the g…
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which of the following is a characteristic of a totalitarian (unlimited…
C. Control over most aspects of citizens’ lives ### Second Question (Government protecting individual rights)
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question 8 of 24 step 1 of 1 use the elementary properties of logarithm…
8
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what is the main difference between limited and unlimited governments? …
C. Limited governments have restrictions on their power while unlimited governments do not ### Second Question (Which of the following is...):
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what is the main characteristic of a free enterprise system? * 1 point …
C. Individuals can own businesses and make economic decisions ### Second Question (Culture Region)
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a line goes through the points (-2, 4) and (6, 20). for numbers 6a - 6c…
TRUE ### 6b
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4 mark for review the function g is defined by $g(x) = \\sqrt{8x + 1}$.…
C. 5
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for the polynomial function f(x) = -2x⁴ - 6x³, answer the parts a throu…
a. D. The graph of \(f(x)\) falls to the left and falls to the right. b. The x - intercepts are \(-3,0\). At \(x=-3\) and \(x = 0\) the graph crosses the x - axis (since their mul…
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determine whether the series converges or diverges. state the test you …
s: (a) The series converges (by p - series test with \(p=\frac{3}{2}>1\)) (b) The series converges (by p - series test with \(p = 4>1\) after substitution) (c) The series diverges…
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question 5 of 24 step 3 of 3 consider the following rational function. …
Four ordered pairs are \( (0, \frac{5}{4}) \), \( (1, \frac{7}{3}) \), \( (2, \frac{9}{2}) \), \( (3, 11) \) (other valid pairs can also be used, e.g., \( x = 5 \): \( f(5)=\frac{…
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consider the following rational function.\\( f(x) = \\frac{-2x - 5}{x -…
\( y = -2 \)