51 Find the equation of the line tangent to the graph of fx

51.) Find the equation of the line tangent to the graph of

f(x) = 2(x^3) - 9(x^2) + 2x - 8 at x = 2.

y =

52.) (a) Find the equation of the tangent line to f(x) = 3(x^3) at the point where x = 2

y =

(b) Graph the tangent line and the function on the same axes. If the tangent line is used to estimate values of the function near x = 2, will the estimates be overestimates or underestimates?


The estimates will be __________.

53.)

Find the equation of the line tangent to the graph of f(t) = 6t - (t^2)

at t = 4.

Sketch the graph of f(t) and the tangent line on the same axes.

The equation of the tangent line is y =

53.) Find the equation of the line tangent to the graph of f(t) = 6t - (t^2) at t = 4. Sketch the graph of f(t) and the tangent line on the same axes.

Solution

1.

y=2x^3 - 9x^2 +2x-8

so, y\'= 6x^2 -18x +2

tangent at x=2, so y\'=-10 and y=-24

so line passing throung (2,-24) and slope of -10

so, -24 = -10*2 +c hence c=-4

so y= -10x -4.

2.

y=3x^3

so, y\'= 9x^2

tangent at x=2, so y\'=36 and y=24

so line passing throung (2,24) and slope of 36

so, 24 = 36*2 +c hence c=-48

so y= 36x -48.

b.

here y\' = 9x^2 is increasing function in 0 to infinity.

So, this tangent line is below the graph.

So, this limit is under estimate the value.

lim x t 2 (3x^3) = 24 but when near to 2, it will under estimate.

3.

f(t) = 4t - t^2

f\'(t) = 4-2t

f(3) = -2

f\'(3) = 3

passing from ( 3,3) and slope of -2 line is

3 = 3*(-2) + c hence c=9

so, y= -2x + 9.