you will explore some functions and theirinverses together with their derivatives and tangent line approximationsat specified points. Perform the following steps using your CAS:a. Plot the function y = ƒ(x) together with its derivative overthe given interval. Explain why you know that ƒ is one-to-oneover the interval.b. Solve the equation y = ƒ(x) for x as a function of y, andname the resulting inverse function g.c. Find an equation for the tangent line to ƒ at the specifiedpoint (x0, ƒ(x0)).d. Find an equation for the tangent line to g at the point(ƒ(x0), x0) located symmetrically across the 45° line y = x(which is the graph of the identity function). Use Theorem 3to find the slope of this tangent line.e. Plot the functions ƒ and g, the identity, the two tangentlines, and the line segment joining the points (x0, ƒ(x0)) and(ƒ(x0), x0). Discuss the symmetries you see across the maindiagonal (the line y = x). y = x3/x2 + 1, -1≤ x≤ 1, x0 = 1/2
you will explore some functions and their
inverses together with their derivatives and tangent line approximations
at specified points. Perform the following steps using your CAS:
a. Plot the function y = ƒ(x) together with its derivative over
the given interval. Explain why you know that ƒ is one-to-one
over the interval.
b. Solve the equation y = ƒ(x) for x as a function of y, and
name the resulting inverse function g.
c. Find an equation for the tangent line to ƒ at the specified
point (x0, ƒ(x0)).
d. Find an equation for the tangent line to g at the point
(ƒ(x0), x0) located symmetrically across the 45° line y = x
(which is the graph of the identity function). Use Theorem 3
to find the slope of this tangent line.
e. Plot the functions ƒ and g, the identity, the two tangent
lines, and the line segment joining the points (x0, ƒ(x0)) and
(ƒ(x0), x0). Discuss the symmetries you see across the main
diagonal (the line y = x). y = x3/x2 + 1, -1≤ x≤ 1, x0 = 1/2
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