Exercises

01

Slope of a tangent line

Find the slope of the tangent line to the given function at the given $x$-value by calculating the limit:

• (a) $f(x)=8-3x$, at $x=4$
• (b) $f(x)=\sqrt{9-x}$, at $x=5$

02

Equation of tangent line

Find the equation of the tangent line to the given function at the given $x$-value using the slope calculations of the previous exercise:

• (a) $f(x)=8-3x$, at $x=4$
• (b) $f(x)=\sqrt{9-x}$, at $x=5$

03

Equation of tangent line - expanding a cubic

By computing the limit definition of derivative, find the equation of the tangent line to the function $f(x)=x^3-12x$ at $x=-3$.

04

Finding $y$-intercept of tangent line as a limit

Compute the $y$-intercept of the tangent line to $y=x^2$ at $x=1$ using a limit that is not (quite) the derivative.

Method: find a formula for the $y$-intercept ‘$b$’ of the secant line passing through the two points $(1,1)$ and $(1+h,(1+h{)}^2)$. Find the limit of this $y$-intercept value as $h\to 0$.