How to find the equation of a tangent line going through a point?
A tangent line is a line that touches a graph in one local point. To find tangent line to a function, we find the derivative of the given function first which gives the slope of the function. We plug in the x value of the point to find the slope of the line. And use the point-slope equation to finally get the equation. For example, let us assume the function is a parabola. $$ y=x^2/2 $$ and the point is $$(-3,9/2)$$. So, $$ f(x)=x^2/2$$. Taking the derivative, $$f' (x)=2x/2=x$$ Plugging in the x coordinate $$ f'(x)=-3$$ Using the equation, $$ y-y_0=m(x-x_0 )$$ $$y-9/2=-3(x+3)$$ $$6x+2y=-9$$ This is the equation of tangent line to $$y=x^2/2$$ at$$ (-3,9/2)$$
How does a suspension bridge like Golden Gate bridge work?
A suspension bridge has five important load carrying components: the deck, the towers, the cables, the suspenders and anchorage blocks. The deck carries the vehicle loads which is connected to the cables with the suspenders. The suspenders carry load in tension and transfers it to the cables which also remain under tension. The cables transfer this force to the towers and to the anchorage blocks. The force on towers acts as compressive force which the towers then transfer to the foundation completing the load path of suspension bridge.
What is the difference between auto-correlation and cross-correlation?
Correlation is a statistical technique that measures how two variables are related or how strongly they are related. It is measured by correlation coefficient that ranges from -1 to 1. For example, if X and Y are two variables with correlation coefficient 1 it means when X increases Y also increases, with -1 means if X increases Y decreases and 0 means the variables do not have any relation. In statistics, autocorrelation refers to the correlation between the values of the same variables. For example, in a time series, autocorrelation measures the similarity of the series with its own displaced version. In other words, it will be the correlation of X(t) and X(t+k) where t refers to time and k is shift in time. On the other hand, cross-correlation measures the correlation between two series at a particular shift. For example the correlation between X(t+k) and Y(t+k) will be called cross-correlation of X and Y series at shift k.