Tutor profile: Maleah C.
Find the derivative of the following expression at x = 3 (4/3)x^3 + sinx - ln(x^2)
Using the sum rule we know that d/dx ((4/3)x^3 + sinx - ln(6x)) = d/dx ((4/3)x^3) + d/dx (sinx) + d/dx(-ln(x^2)) Let's start with the first term Using the power rule we multiply the exponent down and subtract one from the exponent: d/dx ((4/3)x^3) = 3*(4/3)*x^(3-1) Simplifying, we get 4x^2 Plugging in x = 3 4(3)^2 = 36 Next, we move on to the second term d/dx (sinx) = cosx Plugging in x = 3 cos(3) Then, we move on to the third term d/dx(-ln(x^2)) This involves the chain rule Recall that d/dx(ln(g(x)) = 1/g(x) * g'(x) Using this special case of chain rule, we get -1/x^2 * (2x) Simplifying, we get -2/x Pluggin in x = 3 -2/3 Now, we add all the terms together 36 + cos(3) - 2/3 Simplifying we get 106/3 + cos(3)
Sally enjoys riding her bike around her neighborhood with her friends. One day, her and her friends decided to have a race. Sally accelerated from rest to 17.8 m/s/s in 1.56 seconds. She then stayed at this speed for 9.47 seconds. Realizing one of her friends got hurt, she quickly stops in 2.79 seconds. What is Sally's average speed for this motion?
There are 3 stages in this problem: Sally accelerating (1.56s), Sally staying at the same speed (9.74s), and Sally stopping (2.79s) First, we want to calculate the distance Sally travels in first stage (0 to 1.56 seconds) using the equation: d1 = do + Vo*t1 + (1/2)*a1*t1^2 We know that Vo = 0 m/s and do = 0 m because she starts from rest so the equation becomes: d1 = (1/2)a1t1^2 d1 = (1/2)*17.8*(1.56)^2 = 21.66 m Next, we want to calculate Sally's speed during the second stage Using the equation: V1 = Vo + at, we can find the velocity Sally reaches at 1.56 s V1 = 0 + 17.8*(1.56) = 27.77 m/s Using the equation d2 = v*t2, we can find the distance traveled in the second stage d2 = 27.77*(9.47) = 262.96 m Then, we want to calculate Sally's distance traveled in the third stage using the equation: d3 = (Vf+V1)/2*t3 We know that Vf = 0 m/s so our equation becomes: d3 = (V1*t3)/2 d3 = (27.77*2.79)/2 Lastly, we add up the distances traveled in each stage and divide by the total time traveled to get the average velocity Vavg = (d1 + d2 + d3)/t_total = (21.66 + 262.96 + 38.74)/(1.56 + 9.74 +2.79) = 23.4 m/s
Solve for x for the following expression: x - 3x + 7 = 3 + 4x
Combine like terms on each side of the equation -2x + 7 = 3 + 4x Add 2x to both sides 7 = 3 + 4x + 2x Subtract 3 from both sides 7 - 3 = 4x + 2x Combine like terms 4 = 6x Divide both sides by 6 to isolate x x = 4/6 Simplify fraction x = 2/3