Tutor profile: Emilee O.
Find the value of x: $$2x +3= 4$$
$$x= 1/2$$ To find the value of x, we need to solve for x. $$2x+3=4$$ - We need to get x all alone on one side. We do this by subtracting 3 from the left side $$2x= 4-3$$ - Simplify $$2x= 1$$ - Now we need to divide by 2 to get x alone on the left side. $$x= 1/2$$ - And here is our final solution
Find the derivative of $$y=x^4 + 3x^2 + sin(2x)$$
$$dy/dx= 4x^3+ 6x + 2cos(2x)$$ To find the derivative of this equation, we need to use the addition rule or derivatives. This means we can take the derivative of each piece. Using the power rule, the derivative of $$x^4$$ is $$4x^3$$ and the derivative of $$3x^2$$ is 6x. The derivative of $$sin (2x)$$ requires us to take the derivative of $$sinx$$ and of the 2x. So this would become $$2cos(2x)$$. The final derivative is $$dy/dx= 4x^3+ 6x + 2cos(2x)$$
Write an equation of a line that goes through (3,1) and (7,3).
$$y = 1/2x - 1/2$$ First, we need to find the slope of the line (m) using the two points. We use the slope formula of Delta y / Delta x. $$m= (3-1)/ (7-3)= 2/4 = 1/2$$ (Note: It does not matter which point you use first. In this case, I used (7,3) to avoid negatives. ) Next, we will plug in the slope into the slope-intercept formula- y=mx+b $$y= (1/2)x + b$$ To find the 'b', we will plug in a point to their respected x and y values and solve for b. This time let's use the point (3,1). $$1=(1/2)(3) + b$$ $$1=(3/2) + b$$ $$1- (3/2) = b$$ $$b= -1/2$$ Lastly, you plug all the values in to get the final equation that goes through both points: $$y = (1/2)x - (1/2)$$
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