Tangent Lines Assignment.sagews

Tangent Lines Assignment

Question 0

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Question 1

Consider the function $f(x)=3x^2-2x+1$.

Part a

Find the slope of the line tangent to $f$ at the point $(1,2)$ using limits.

Part b

Find an equation for this tangent line.

Part c

Graph $f$ and its tangent line on the same axes with $0 < x < 2$.

Question 2

Consider the function $g(x)=e^{-x^2}$. [Caution: $e$ is not a variable, so do not declare it.]

Part a

Find the slope of the line tangent to $g$ at the point $(2,e^{-4})$ using limits.

Part b

Find an equation for this tangent line.

Part c

Graph $g$ and its tangent line on the same axes with $1 < x < 3$.

Question 3

Consider the function $F(x)=\sin(3x)+\cos(2x)$.

Part a

Find the slope of the line tangent to $F$ at the point $(0,1)$ using limits.

Part b

Find an equation for this tangent line.

Part c

Graph $F$ and its tangent line on the same axes with $-1<x<1$.

Question 4

Consider the function $G(x)=2x^3+3x^2-36x+30$.

Part a

Plot a graph of $G(x)$ with $-5\le x \le 5$. Notice that $G(x)$ appears to have relative extrema at $x=-3$ and $x=2$.

Part b

Confirm that $G(x)$ has horizontal tangent lines at $x=-3$ and $x=2$ (i.e., calculate the slope of the tangent line, and see that it is 0).

This material was developed by Aaron Tresham at the University of Hawaii at Hilo and is
licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.