Tangent Line Exercises

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Here we’ll practice finding lines tangent to curves.

Consider $f(x) = x^2$

(a): What is the equation of the secant line connecting the point $(1,f(1))$ to the point $(1.1,f(1.1))$ ?
The equation of this secant line is $y = \answer { 2.1x - 1.1}$
(b): Use the definition of the derivative to find the slope of the tangent line to $f$ at $x=1$ , and use this to find the equation of this tangent line.
The equation of this tangent line is $y = \answer {2x - 1}$

Consider $f(x) = \frac {1}{x^2}$

(a): What is the equation of the secant line connecting the point $(1,f(1))$ to the point $(0.9,f(0.9))$ ?
The equation of this secant line is $y = \answer { -2.34568x+ 3.34568}$
(b): Use the definition of the derivative to find the slope of the tangent line to $f$ at $x=1$ , and use this to find the equation of this tangent line.
The equation of this tangent line is $y = \answer { -2x + 3}$

Consider $f(x) = \frac {x}{2}+\frac {2}{x}$

(a): What is the equation of the secant line connecting the point $(2,f(2))$ to the point $(2.1,f(2.1))$ ?
The equation of this secant line is $y = \answer {0.0238095x+1.95238 }$
(b): Use the definition of the derivative to find the slope of the tangent line to $f$ at $x=2$ , and use this to find the equation of this tangent line.
The equation of this tangent line is $y = \answer {2}$