Please help me solve these problems. I tried, but I can't figure them out.
1) If is a differentiable function of then the slope of the tangent to the curve at the point where [tex]y=1[tex] is..?
2)Find the point on the graph of y=x^(1/2) between (1,1) and (9,3) at whtich the tangent to the graph has the same slop as the line through (1,1) and (9,3)
3) Let f and g be differential functions such that :
f(1) = 4
g(1) = 3
f'(3) = -5
g'(3) = 2
If h(x) = f(g(x)), then h'(1) =
4) Find the derivative and the equation of the tangent line to
5) Find the second derivative of
6) Use the tangent line approximation of y= square root of x at x=16 to find the approximate value of the square root of 17
And also, what does differentiable mean? How can you tell on a graph if a point is differentiable or not?
Thanks for the help
then, find the derivative of y = x^(1/2) and set it equal to the value you found above. solve for x, this will give you the x-value of the point you are after. then plug this x-value into the original function to solve for the corresponding y-value, and then you'll have your point
i will use y' to mean dy/dx to save on typing
differentiating implicitly we get:
...........note that i used the chain rule for the first term
now, solve for y' and plug in the required x and y-values
to find the tangent line, we need to know at what point. when you have the point, use the point-slope form:
where is the slope of the tangent line at the point and is a point the line passes through
just solve for y
here is the value for which we want to find for, but we can't because it does not have a nice answer, and is a point close to for which we do know the exact value of the function.
here , and
recall how you defined the derivative. we started out with the slope formula for the equation of a line. thus, in layman's terms, a function is differentiable at a point, if it is approximately linear at that point. that is, if we zoom in close to a point, the function looks like a straight line. what this translates to is the following.How can you tell on a graph if a point is differentiable or not?
- we cannot find the derivative at points where a function takes a sharp turn, e.q. y = |x| at x = 0
- we cannot find the derivative of a function where it is discontinuous