Subsections

• Purpose
• Exercises

Derivatives

Purpose

The purpose of this lab is to teach you how to use Maple commands for computing derivatives.Use last week's lab to help you with the commands for this lab.

Exercises

1. Enter the expression $$\vert x^2-4\vert^{\frac{1}{3}}$$,

   a)

   Compute the derivative and label the derivative using a variable name.

   b)

   Evaluate the derivative at $x=-1$, $x=0$, and $x=2$.

   c)

   Plot the expression over the interval $-5 \leq x \leq 5$. Using this plot, can you explain why the expression was not differentiable at one of the $x$ values given above?

2. Find the equation of the line tangent to the function $$f(x)=\cos(x)+\sin(x)-x^3+12x-1$$ at $x=3$. When calculating the derivative at a point, use the $D$ command. Include a plot of the function and the tangent line on the same graph over the interval $0 \leq x \leq 5$.

3. For the same function $$f(x)=\cos(x)+\sin(x)-x^3+12x-1$$,

   a)

   Plot $f(x)$ over the interval $-5 \leq x \leq 5$ and state how many horizontal tangent lines to the graph there are.

   b)

   Plot the derivative of $f$ over the same interval. Explain how this supports your answer above.

   c)

   Labeling each fsolve command, find the $x$ values where a horizontal tangent line is located. Find the corresponding $y$ values. State in text all points on the graph of $f(x)$ where the tangent line is horizontal.

4. The tangent line to a function at a particular value of x intersects the graph of the function at least once, at the point of tangency. However, the tangent line may intersect the graph at other points. In this problem, we investigate whether the tangent line at one point can also be tangent to the graph at another point. For example, consider the function
   
   $$g(x) = (x^2-1)^2 +\frac{x}{2}$$
   
   
   Find the point(s) where the tangent line at $x=-1$ is also tangent to the graph.