Final answer:
To express the complex number 5-7j in polar form, calculate its magnitude and angle. The angles between vectors are found using the dot product. Differentiate and integrate each term of the given functions with respect to x by applying the appropriate rules.
Step-by-step explanation:
The complex number 5-7j can be expressed in polar form by calculating its magnitude and angle. The magnitude is √(52 + (-7)2), and the angle θ can be found using the arctangent function: θ = atan2(-7, 5). To find the angle between two vectors, we use the dot product formula which relates the dot product to the magnitude of the vectors and the cosine of the angle: cos(θ) = (a · b) / (|a| |b|). Differentiation and integration of functions with respect to x involve applying the respective rules to each term.
Example of differentiation of y:
· For 2x9, the derivative is 18x8.
· For -3x4, it is -12x3.
· The derivative of -5x is -5.
· The derivative of 3cos(5x) is -15sin(5x).
· For -e-x, the derivative is e-x.
Example of integration of y:
· The integral of 4x3 is x4.
· The integral of 3x5 is ⅓x6.
· For 8sin(4x), it is -2cos(4x).
· The integral of 2cos(2x) is sin(2x).
· For -e3x, the integral is -⅓e3x.