Use Logarithmic Differentiation to Find the Derivative of the Function y = x^x
To find the derivative of the function y = x^x, we can use logarithmic differentiation. This technique is especially useful when dealing with functions where the variable appears in both the base and the exponent. First, we take the natural logarithm of both sides: ln(y) = ln(x^x) Using the properties of logarithms, we can simplify […]
Which input value produces the same output value for the two functions on the graph x^3 and x^2 + x – 3?
To find the input value that produces the same output for the functions f(x) = x^3 and g(x) = x^2 + x – 3, we need to set the two functions equal to each other: x^3 = x^2 + x – 3 Rearranging the equation gives us: x^3 – x^2 – x + 3 = […]
What is the remainder when x³ + 1 is divided by x² + x + 1?
To find the remainder when dividing a polynomial, we can use polynomial long division. In this case, we need to divide x³ + 1 by x² + x + 1. 1. First, we set up the long division. We divide the leading term of x³ (which is x³) by the leading term of x² (which […]
Convert 1450 mm to meters and express the answer to the correct number of significant digits
To convert millimeters to meters, we can use the conversion factor where 1 meter is equal to 1000 millimeters. So, to convert 1450 mm to meters, we divide the number of millimeters by 1000: Calculation: 1450 mm ÷ 1000 = 1.450 m Now, to express 1.450 meters with the correct number of significant digits, we […]
Find the midpoint of the segment with the given endpoints D(1, 2) and E(3, 6)
To find the midpoint of a segment with given endpoints, we use the midpoint formula: Midpoint = ( (x1 + x2) / 2 , (y1 + y2) / 2 ) For our points D(1, 2) and E(3, 6): x1 = 1 y1 = 2 x2 = 3 y2 = 6 Now plug these values into […]
How can you estimate 22 of 78 to check if your answer is reasonable?
To estimate 22 of 78, you can round the numbers to make the math easier. Start by rounding 78 to a more manageable number, such as 80. Then, you can calculate 20% of 80, which is a simpler way to approach the problem. Here’s how it works: 20% of 80 is calculated as follows: First, […]
Convert the angle 613845 to decimal degrees and round to the nearest hundredth of a degree
To convert the angle 613845 to decimal degrees, we first need to understand the format of this number. The number represents an angle in a degree-minute-second format (DMS), where: The first two digits (61) represent degrees. The next two digits (38) represent minutes. The last two digits (45) represent seconds. Now, we can perform the […]
If cos(8) = √2/2 and 3π/2 < 8 < 2π, what are the values of sin(8) and tan(8)?
To find the values of sin(8) and tan(8), we can start by analyzing the given information. We know that cos(8) = √2/2, which corresponds to specific angles in the unit circle. Since √2/2 is the cosine of both π/4 (45 degrees) and 7π/4 (315 degrees), we need to determine which angle 8 is near, given […]
Find 2x² + 2z⁴ + y²x²z⁴ if x = 4, y = 3, and z = 2
To find the expression 2x² + 2z⁴ + y²x²z⁴ given the values x = 4, y = 3, and z = 2, we can substitute these values into the equation step by step. First, we calculate each variable separately: x² = 4² = 16 z⁴ = 2⁴ = 16 y² = 3² = 9 Now, […]
How to Solve the Quadratic Equation Using the Square Root Property: 2x² – 52 = 49
To solve the quadratic equation 2x² – 52 = 49 using the square root property, we first need to rearrange the equation to isolate the x² term. 1. Start by moving 49 to the left side of the equation: 2x² – 52 – 49 = 0 2x² – 101 = 0 2. Next, we add […]