Mathematics Placement/Pretest 2 Robotics

The given equation is θ = arctan ((2x + 1) / 4). This equation represents an inverse function, where θ is the angle whose tangent is equal to ((2x + 1) / 4). The inverse function allows us to write θ as a function of x, giving us the correct answer.

The function y = cos (x - (π/2)) is a shifted cosine function. The original cosine function has a period of 2π and starts at its maximum value of 1 when x = 0. By subtracting π/2 from x, the function is shifted π/2 units to the right. This means that the maximum value of the function occurs when x = π/2. Graph C is the only graph that starts at its maximum value at x = π/2 and has a period of 2π, so it is the correct graph for the given function.

When cosine (cos) θ is less than 0, it means that θ lies in either Quadrant II or Quadrant III. When cosecant (csc) θ is less than 0, it means that θ lies in either Quadrant III or Quadrant IV. Since both conditions are satisfied, θ must lie in Quadrant III where both cos θ and csc θ are negative.

The given equation x = 4 tan θ can be rewritten as sec θ = x/4. Since the question asks for the expression in terms of θ, we can conclude that the correct answer is 4secθ.

The given expression can be condensed to log (((7x)^(1/3)) / y) by using the properties of logarithms. The expression 1/3[log x + log 7] can be simplified to log ((7x)^(1/3)), and [log y] can be written as -log y. Combining these terms, we get log (((7x)^(1/3)) / y).

The given quadratic function, f(x) = -x^2 + 2x + 8, can be rewritten in standard form by completing the square. To do this, we need to rewrite the quadratic in the form (x - h)^2 + k, where (h, k) represents the vertex of the parabola. By completing the square, we get f(x) = -(x - 1)^2 + 9, which matches the given answer.

The polynomial f(x) can be factored using the given factors (x - 1) and (x + 4). By multiplying these factors with the remaining factor (x + 2), we can obtain the fully factored form of f(x) as (x - 1)(x + 4)(x + 2).

The x-intercept of a function is the point at which the function intersects the x-axis. In this case, the function is y = 3 + log3 X. To find the x-intercept, we need to set y equal to zero and solve for X. Setting y = 0, we get 0 = 3 + log3 X. Subtracting 3 from both sides gives log3 X = -3. Exponentiating both sides with base 3 gives X = 3^(-3), which simplifies to X = 1/27. Therefore, the x-intercept of the function is 1/27.

The vertical asymptote of a function occurs when the function approaches infinity or negative infinity as x approaches a certain value. In the given function f(x) = 2 + log(x + 3), the logarithm term is only defined for positive values of x. Therefore, as x approaches -3 from the right side, the function approaches negative infinity. Hence, x = -3 is the vertical asymptote of the function.

The correct answer is x-intercept: (0, 0); y-intercept: (0, 0). This means that the function does not have any x-intercepts, as the numerator of the function is always positive and the denominator is always positive. Therefore, the function never equals zero. The y-intercept is (0, 0) because when x is equal to 0, the function evaluates to 0.

The given expression can be simplified using the properties of logarithms. The property states that when subtracting logarithms with the same base, we can divide the numbers inside the logarithms. Applying this property, we have log4 12 - log4 3 = log4 (12/3) = log4 4 = 1. Therefore, the exact value of log4 12 - log4 3 is 1.

The given expression can be expanded as a sum of logarithms. It can be written as log 6 + 5(log x) - log y.

The given question asks for the exact value of the fraction 3/5. The numerator represents the number of equal parts being considered, which is 3 in this case. The denominator represents the total number of equal parts in the whole, which is 5 in this case. Therefore, the fraction 3/5 represents three out of five equal parts, which can also be expressed as a decimal value of 0.6 or a percentage of 60%.

Given that cot θ = 7/24 and csc θ > 0, we can use the identity csc θ = 1/sin θ. Since csc θ is positive, sin θ must also be positive.

We can find sin θ by using the Pythagorean identity, sin^2 θ + cos^2 θ = 1. Since cot θ = 7/24, we can find cos θ by using the identity cot θ = cos θ/sin θ. Solving for cos θ, we get cos θ = 24/7.

Now, we can substitute the values of sin θ and cos θ into the Pythagorean identity and solve for sin θ. Plugging in sin θ = x, we have x^2 + (24/7)^2 = 1. Solving this equation, we find that x = 24/25.

Therefore, the exact value of sin θ is 24/25.

The point (7, 24) lies on the terminal side of an angle in standard position. To determine the exact value of tan θ, we can use the coordinates of the point. Since tan θ is equal to the ratio of the y-coordinate to the x-coordinate, we have tan θ = 24/7.

To solve the equation (1/3)7x-1 = 27, we can use the One-to-One Property which states that if two exponential expressions with the same base are equal, then their exponents must be equal as well.

First, we can simplify the left side of the equation by multiplying both sides by 3 to get rid of the fraction. This gives us 7x-1 = 81.

Next, we can add 1 to both sides of the equation to isolate the variable. This gives us 7x = 82.

Finally, we can divide both sides by 7 to solve for x. This gives us x = 82/7, which simplifies to -2/7.

Therefore, the correct answer is -2/7.

The given expression is cot θ - 1. The equivalent expression to this is cot θ itself. This is because cot θ is equal to 1/tan θ, so subtracting 1 from cot θ gives the same result as the original expression. Therefore, the correct answer is cot θ.

The given function is a rational function, which means it can have both slant and vertical asymptotes. To find the slant asymptote, we divide the numerator by the denominator using long division or synthetic division. The quotient of the division will give us the equation of the slant asymptote, which in this case is y = x - 2. To find the vertical asymptote, we set the denominator equal to zero and solve for x. In this case, the denominator is x + 5, so the vertical asymptote is x = -5.

The given expression, F(x) = x^3 - 3x^2 + 4x - 12, can be factored as (x - 3)(x + 2i)(x - 2i). This is because the expression can be rewritten as (x - 3)(x^2 + 4), and further simplified as (x - 3)(x + 2i)(x - 2i) by using the difference of squares formula.

The cosecant function (csc x) is the reciprocal of the sine function (sin x). Since sin x is equal to 1/2, the reciprocal of 1/2 is 2. Therefore, csc x is equal to 2.

The equation 2 cos x - 1 = 0 can be solved by isolating the cosine term and then finding the values of x that satisfy the equation. By adding 1 to both sides and dividing by 2, we get cos x = 1/2. This means that x is an angle whose cosine is equal to 1/2. The values of x that satisfy this condition are (π/3) + 2nπ and (5π/3) + 2nπ, where n is an integer.

The formula for continuous compound interest is given by A = P*e^(rt), where A is the final amount, P is the principal amount, r is the interest rate, and t is the time in years. In this case, we want to find the time it takes for the investment to double, so we set A = 2P and solve for t. Plugging in the values, we get 2 = e^(0.08t). Taking the natural logarithm of both sides, we get ln(2) = 0.08t. Solving for t, we find t = ln(2)/0.08 ≈ 8.66 years. Therefore, it will take approximately 8.66 years to double the investment.

The given function is f(x) = 2 / (x - 3). To determine the horizontal asymptote, we need to consider the behavior of the function as x approaches positive or negative infinity. As x becomes very large or very small, the denominator (x - 3) becomes much larger than the numerator (2), resulting in the function approaching 0. Therefore, the horizontal asymptote is y = 0. To determine the vertical asymptote, we need to find the values of x that make the denominator equal to 0. In this case, x = 3 makes the denominator 0, so the vertical asymptote is x = 3.

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The vertex of a quadratic function in the form f(x) = ax^2 + bx + c is given by the coordinates (-b/2a, f(-b/2a)). In this case, the quadratic function is f(x) = x^2 - 3x + 13/4. Comparing it to the standard form, we can see that a = 1, b = -3, and c = 13/4. Plugging these values into the formula, we get the vertex coordinates as (-(-3)/(2*1), f(-(-3)/(2*1))) = (-3/2, f(3/2)). Evaluating the function at x = 3/2, we find f(3/2) = (3/2)^2 - 3(3/2) + 13/4 = 9/4 - 9/2 + 13/4 = 1. Therefore, the vertex of the graph is (-3/2, 1).

The given answer lists all the values of x that satisfy the equation in the interval [0, 2π). These values are 0, π/4, 3π/4, π, 5π/4, and 7π/4.