微积分
极限、导数、最优化与函数性质。
-
the table shows three unique functions. | x | f(x) | g(x) | h(x) | |---…
- \( g(x) \) has the greatest maximum: Correct (checked values). - \( f(x) \) and \( h(x) \) have \( x \)-intercepts: Correct (both \( y=0 \) at \( x=1 \)). - The range of \( g(x)…
-
10. use the graph of the rational function to complete the following st…
\( -1 \) (assuming horizontal asymptote is \( y = -1 \) for the first graph) ### Problem 11:
-
if y = -cos x, what x-value corresponds to a y-value of \\(\\frac{1}{2}…
\(\frac{4}{3}\)
-
9. use descartess rule of signs to determine the possible numbers of po…
Possible number of positive real zeros: \( 2, 0 \) Possible number of negative real zeros: \( 2, 0 \)
-
graph the following function. $y = 3 \\cdot 4^x$ show your work here
To graph \(y = 3\cdot4^{x}\): 1. Plot the y - intercept \((0,3)\) and the point \((1,12)\). 2. Draw a smooth exponential growth curve through these points, with the x - axis (\(y …
-
14. find all radian solutions to the equation (tan(3x))² = 1.
The solutions are \(x=\frac{\pi}{12}+\frac{n\pi}{6}\), where \(n\) is any integer (\(n\in\mathbb{Z}\)).
-
y = 1·3^x calculate y by substituting x = 0 into y = 1·3^x.
\( y = 1 \)
-
calculate y by substituting x = 0 into y = 1·3^x.
\( y = 1 \)
-
graphing polynomial functions which statement describes the graph of ( …
The second option (The graph touches the \( x \)-axis at \( x = 0 \) and crosses the \( x \)-axis at \( x = -5 \))
-
what is the end behavior of the graph of the polynomial function $f(x) …
The first option (As \( x \to -\infty \), \( y \to \infty \) and as \( x \to \infty \), \( y \to \infty \))
-
at which root does the graph of $f(x) = (x - 5)^3(x + 2)^2$ touch the $…
-2
-
what is the end behavior of the graph of the polynomial function $f(x) …
The fourth option: As \( x \to -\infty \), \( y \to \infty \) and as \( x \to \infty \), \( y \to -\infty \) (Wait, correction: Wait, for a cubic function \( f(x)=ax^3+bx^2+cx + d…
-
find the slope of the line that passes between (-3, -4) and (5, -12) sl…
\(-1\)
-
which statement describes the graph of $f(x) = -4x^3 - 28x^2 - 32x + 64…
The graph crosses the \( x \)-axis at \( x = -4 \) and touches the \( x \)-axis at \( x = 1 \). (First option)
-
what is the domain of the function graphed below? a. real numbers great…
C. all real numbers greater than or equal to -3
-
at which root does the graph of $f(x) = (x + 4)^6(x + 7)^5$ cross the $…
-7
-
graph the following function. $y = 5 \\cdot 2^x$ show your work here (t…
To graph \( y = 5 \cdot 2^x \): 1. Plot key points: \( (0, 5) \), \( (1, 10) \), \( (-1, 2.5) \). 2. Draw a smooth curve through these points, showing exponential growth (increasi…
-
let a and b be real numbers, where a ≠ b ≠ 0. which of the following fu…
$f(x) = x(x - a)(x - b)^2$
-
consider the function f(x). $f(x) = x^4 - 7x^3 + 6x^2 + 32x - 32$ which…
A. The graph includes the point \((0, -32)\), B. The graph crosses the \(x\)-axis at the point \((4, 0)\), C. As \(x\) approaches positive infinity, the value of \(f(x)\) increase…
-
ntifying the key features of a function from its graph what intervals w…
$(-\infty, -2)$, $(-2, 4)$, and $(4, \infty )$
-
identifying the key features of a function from its graph of the zeroes…
None
-
find the equation of the exponential function. | x | f(x) | | --- | ---…
\( f(x) = 3 \cdot 2^x \)
-
modeling real life a rock climber averages $12\\frac{3}{5}$ feet per mi…
378
-
escribing the end behavior of a function complete the statements that d…
The graph touches, but does not cross, the \( x \)-axis at \( x = 4 \). The graph of the function crosses the \( x \)-axis at \( x = 0 \).
-
identifying the key features of a polynomial function complete the stat…
- $-3$ with multiplicity 1 - $3$ with multiplicity 1 - $0$ with multiplicity 3
-
select the best description of the following exponential equation. $f(x…
Growing by 843% with every unit of \( x \) (the corresponding option, e.g., if the first option is "Growing by 843% with every unit of \( x \)", then the answer is that option).
-
drag and drop the correct numbers into each box in the table below to i…
For $\log_{3}3^{5}$: $5$ For $\log_{10}(0.00001)$: $-5$ For $\ln e^{-3}$: $-3$
-
select the best description of the following exponential equation. $f(x…
Growing by 504% with every unit of \( x \) (the option: Growing by 504% with every unit of \( x \))
-
austin jumped off a cliff into a river. the following graph of a quadra…
3 (assuming the option is "3" as the correct time when he lands in the river, since that's when h(t)=0, the positive root of the quadratic representing his height over time.)
-
identifying multiplicity of roots graphically the graph of f(x) is give…
3
-
identifying multiplicity of roots graphically the graph of f(x) is give…
-1, 3
-
5. g(x) = \\frac{x + 2}{x - 5} graph (with a grid graph below)
To graph \( g(x)=\frac{x + 2}{x - 5} \), we use the vertical asymptote \( x = 5 \), horizontal asymptote \( y = 1 \), x - intercept \( (-2,0) \), y - intercept \( (0,-\frac{2}{5})…
-
multiple choice 1 point an angle has an initial side of fg and a termin…
3.927
-
2. complete this information about the pair of cards that includes a1 w…
- Exact value for \( f(3) \): \( \frac{48}{7} \) - Solution for \( f(x) = -1 \): \( x = -2 \)
-
5 multiple choice 1 point consider the function f(x) = 4 sin (2x) + 3. …
A. 4
-
4. $m(x) = \\frac{-3}{x} - 4$
To graph \( m(x)=\frac{-3}{x}-4 \): 1. Draw the vertical asymptote \( x = 0 \) (the \( y \)-axis) and the horizontal asymptote \( y=-4 \) (a horizontal line 4 units below the \( x…
-
a metal ball attached to a spring is bouncing up and down. at 0.5 secon…
\( h(t) = 40\sin(\pi t) \) (the third option)
-
1 multiple choice 1 point which function does not have the set of all r…
\( f(x)=\frac{x - 1}{x - 3} \) (the third option: \( f(x)=\frac{x - 1}{x - 3} \))
-
3. $k(x) = \\frac{1}{x - 3} + 5$
To graph $k(x) = \frac{1}{x - 3} + 5$: 1. **Asymptotes**: Vertical at $x = 3$, horizontal at $y = 5$. 2. **Transformations**: Shift $y = \frac{1}{x}$ 3 units right, 5 units up. 3.…
-
identifying key features for graphing polynomial functions given $f(x) …
The second option (the middle - left option: \( f(x)\to-\infty \) as \( x\to-\infty \); \( f(x)\to+\infty \) as \( x\to+\infty \))
-
khloe reaches the end of a hiking trail and hikes back to her car at a …
-3
-
a helicopter takes off from the top of a building. the helicopter’s hei…
- For 0: Neither - For 5: Neither - For 10: Slope - For 50: y - intercept - For 100: Neither (In the table, for each value: - 0: Select Neither - 5: Select Neither - 10: Select Sl…
-
florencia and eli each dropped a football from different heights. the h…
Eli
-
pihu and angelica are each designing a suspension bridge. the height (i…
Pihu's design (since the minimum height of Pihu's cable is approximately 25.3 feet and Angelica's is around 15 feet).
-
in exercises 1 and 2, graph the function. compare the graph with the gr…
(for the comparison part, as the graphing is a visual task): - For \( g(x)=\frac{0.25}{x} \): The graph of \( g(x) \) is a vertical compression of the graph of \( f(x)=\frac{1}{x}…
-
b) $g(x) = \\frac{1}{2} \\cdot 4^x$ left: $\\lim\\limits_{x \\to -\\inf…
- Left Limit: $\boldsymbol{0}$ - Right Limit: $\boldsymbol{\infty}$ - Increasing or Decreasing: $\boldsymbol{\text{Increasing}}$ - Concave Up or Concave Down: $\boldsymbol{\text{C…
-
a weather balloon was released from the top of a 500-foot-tall building…
The rate of change of the weather balloon is 15 ft/s, and the initial value is 500 feet.
-
1 exploration: identifying graphs of rational functions (continued) c. …
The graph labeled E (the graph in the bottom - left, labeled E)
-
what is the rate of change of the function passing through the points (…
\(\frac{4}{3}\) (The option with \(\frac{4}{3}\))
-
a dance school offers classes in ballet folklórico. there is a one - ti…
\( y = 8x + 12 \)
-
how would you limit the domain to make this function one-to-one and sti…
\(-3\)
-
which polynomial function has a leading coefficient of 3 and roots -4, …
\( f(x) = 3(x + 4)(x - i)(x + i)(x - 2) \)
-
(03.02 lc) read the following lines of dialogue and answer the question…
D. unlimited authority, complete freedom
-
3. if $f(2) = 4$, then find $limlimits_{delta x \to 0} \frac{f(2 - 2del…
\(-8\)
-
1 exploration: identifying graphs of rational functions (continued) c. …
- **Intercepts**: - x - intercept: \( (a,0) \) - y - intercept: \( (0,\frac{a}{b}) \) (for \( b eq0 \)) - **Asymptotes**: - Vertical Asymptote: \( x = b \) - Horizontal Asymptote…
-
essential question what are some of the characteristics of the grapt a …
If we are matching function d (\(g(x)=\frac{x - 2}{x + 1}\)) with graph A, then the answer is that function \(g(x)=\frac{x - 2}{x + 1}\) matches graph A. The reasoning is that \(g…
-
rate of change: -100 point on the graph: (6, 900) solve for b to find t…
\(1500\)
-
summary sheet #6: exponential and logarithmic functions exponential fun…
| Characteristic | Exponential Function (\( y = 3^x \)) | Logarithmic Function (\( y=\log_3 x \)) | |----------------|--------------------------------------|----------------------…
-
without graphing, identify the vertex, axis of symmetry, and transforma…
F. The function is translated 4 units to the right and 5 units down.
-
6. simplify: (9 + 4 - 2) + (4 • 2)
19
-
rate of change: -100 y = -100x + b the initial value is the amount of a…
\((6, 900)\)
-
without graphing, identify the vertex, axis of symmetry, and transforma…
D. The function is translated 1 unit to the left and 2 units up.
-
question graph the following function on the axes provided. $f(x) = \\b…
(Graphing steps as above: first segment from open \((-4, 3)\) to closed \((-1, -6)\), second segment from open \((-1, -2)\) to closed \((5, -8)\))
-
choosing the equation of a polynomial function which of the following p…
\(3x^{3}+3x^{2}-6x\)
-
find the slope of the line: slope = \\frac{change in y}{change in x} sl…
\(-\frac{1}{2}\)
-
31. which equation is correct? a (-14 - (-13.5) = 0.5) b (-14 - (13.5) …
5.8 (assuming the expression is $-\frac{1}{2}(3 - 7)-(1.2 - 5)$)
-
while being deflated, a hot - air balloon loses 300 cubic meters of air…
\(-100\)
-
find the horizontal asymptote of the graph of the rational function. $y…
A. \( y=\frac{1}{18} \)
-
use the properties of logarithms to find the value of x in the followin…
For $2 \cdot \log_{9}x = 6$: 729 For $\log_{7}7^8 = x$: 8
-
list all the asymptotes of the following function, then graph the funct…
B. \( x = 3, x = -3, y = 0 \)
-
question 18 use the properties of logarithms to find the value of x in …
64
-
find the horizontal asymptote of the graph of the rational function. $y…
\( y = 0 \)
-
what is the value of the function $y = 3x - 1$ when $x = -1$? -8 -4 2 6
\(-4\) (corresponding to the option with \(-4\))
-
a hot - air balloon is flying at an altitude of it rises at a constant …
Input (x - value): number of minutes Output (y - value): altitude of the balloon The rate of change is: Increases
-
find the vertical asymptotes of the graph of the rational function. $y …
\(-16\)
-
use long division to rewrite the rational function. what are the asympt…
### For Horizontal Asymptote: The function is rewritten as \( f(x) = 4 - \frac{24}{x + 6} \). For a rational function, when the degrees of the numerator and denominator are equal …
-
use long division to rewrite the rational function. what are the asympt…
A. There is a horizontal asymptote defined by the line \( y = 4 \)
-
use long division to rewrite the rational function. what are the asympt…
\(4-\frac{24}{x + 6}\)
-
question 19 of 20 with just a high school education, it is expected tha…
D. annual earnings will be lower and job choices will be more limited.
-
part 3 of 4 use long division to rewrite the rational function. what ar…
### For Horizontal Asymptote: To find the horizontal asymptote of a rational function \( f(x)=\frac{3x}{x + 4}\) (or its rewritten form \( f(x)=3-\frac{12}{x + 4}\)), we analyze t…
-
use long division to rewrite the rational function. what are the asympt…
A. There is a horizontal asymptote defined by the line \( y = 3 \)
-
riting a polynomial function of a graph use the fact that the graph pas…
2
-
1. piper is willing to pay a high premium for their disability insuranc…
C. They receive a low deductible and a high coverage limit
-
writing a polynomial function of a graph write the 4th degree polynomia…
\((x + 3)(x + 1)(x-2)(x - 3)\) (the second option in the list of factors)
-
use long division to rewrite the rational function. what are the asympt…
\(3-\dfrac{12}{x + 4}\)
-
which is the equation of the function in the graph? y = 4x + 8 y = 2x +…
\( y = 2x + 8 \) (Option: \( y = 2x + 8 \))
-
enter your answer in the box. y = \\square x + \\square
\(y = \boldsymbol{10}x+\boldsymbol{10}\)
-
enter your answer in the box. y = \\boxed{} x
3
-
enter your answer in the box. y = \\square x - \\square
The first box is \(-5\) and the second box is \(3\), so \(y=-5x - 3\) (i.e., \(y=\boldsymbol{-5}x-\boldsymbol{3}\))
-
im2301_mvoa integrated math iii semester a_2439883 writing polynomial f…
\(A = \boxed{1}\), \(B = \boxed{4}\), \(C = \boxed{4}\)
-
(c) \\log_{4} \\sqrt5{256} (f) \\log_{2} \\sqrt7{512}
- For part (c): \(\frac{4}{5}\) - For part (f): \(\frac{9}{7}\)
-
the complex conjugates theorem states that a 3 - i must also be a root.…
\((x - 1)[x - (3 + i)][x - (3 - i)]\) (the first option)
-
graph the function. identify the x - and y - intercepts and the asympto…
B. The function has no x - intercept.
-
simplify your answer and write it as a proper fraction, improper fracti…
\(-\frac{1}{2}\)
-
Question was provided via image upload.
a. Vertex: \((-2, 25)\) b. y - intercept: \((0, 21)\) c. The \(\text{maximum}\) value is \(25\) d. x - intercept(s): \((-7, 0)\) and \((3, 0)\) e. Factored form: \(f(x)=-(x + 7)(x…
-
epts and asymptotes of the following rational function. if you get a fr…
Vertical Asymptote: \(x=-5\), Horizontal Asymptote: \(y = - 6\), x - Intercept: \((-6,0)\), y - Intercept: \((0,-\frac{36}{5})\)
-
6. affirmative action refers to a. the u.s. senate approving a presiden…
e. a policy giving special consideration to groups that have been disadvantaged historically. ### Question 7
-
inverse functions coloring activity directions: find the inverse of eac…
(Example of one inverse pair: For \( y = 12x - 3 \), inverse is \( y=\frac{x + 3}{12} \) (H, red); For \( y=-8x + 16 \), inverse is \( y=\frac{x - 16}{-8} \) (A, grey); etc. The p…
-
log\\frac{9x^{7}y}{x^{2}y^{4}} = a\\log3 + b\\log x + c\\log y
$A = 2$, $B = 5$, $C=-3$
-
4. in a video game, the amount of damage a player’s character can do is…
\( D(2)=4 \) ### Part (b)