Tutor profile: Geovani I.

evaluate the following expression, 7x8 + (24/8 x 4)

let us recall the following, PEMDAS; Parentheses Exponents Multiplication Addition Subtraction remember that if you have multiplication and division inside the parentheses, do whatever comes first from left to right. so to evaluate the problem we first have to divide 24 by 8 which will give us the following 7x8+(3x4) now we have to multiply 3 times 4 because they are still in side the parentheses, so we get the following 7x8 + 12 now we multiply 7 times 8 because it is multiplication and then we add the result with 12 and get the final answer of 68 56 + 12 = 68

find the first, second and the derivative of the following function, y = 5x^3

remember, to find the derivative a polynomial we can use the following rule. y = x^a where a is any real number greater than 0 to find the first derivative we use the following rule y'= a X^(a-1) to find the second derivative we take the derivative of the first derivative, y'' = a (a-1) X^(a-2) if we want to get the third derivative, we take the derivative of the second derivative, y''' = a (a-1) (a-2) X^(a-3) let us apply these rules to our original function, first derivative will be y' = 15x^2 + 2x +1 second derivative will be y'' = 30x + 2 third derivative will be y''' = 30

Solve the ordinary differential equation (ODE) dx/dt = 5x-3 for x(t)

let us rearrange the equation with their respective variables, dx/(5x-3) = dt if we integrate both sides we get the following, 1/5 ln(5x-3) = t + C1 let us multiply by 5 across the whole equation to get rid of the fraction on the left side, by doing that we get the following, ln(5x-3) = 5t+5C1 now in order for us to get rid of the natural log (ln) we can take the exponent on both sides of the equation, if we do it we get the following, 5x-3 = e^(5t+5C1) let us not forget that we are finding a solution for x(t), so lets add 3 to both sides of the equation and then divide by 5, by doing so we get the following, x = (1/5)e^(5t+5C1) + (3/5) in order to simplify this, if we recall we can split the exponent e^(5t+5C1) into the following e^(5t) * e^(5C1) , with this we can let C = 1/5 e^(5C1) since our exponential has constants, we will end up getting our final equation, x(t) = Ce^(5t) + 3/5, where C = 1/5 * e^(5C1)