Solutions to Quiz 1

Problem 1

Tell whether each of the following expressions is infinitesimal, moderate, or infinite.

Limit[(e-2)/e^2,e->0]

     -Infinity

Limit[(e^2-1/e)/(e^2+1/e),e->0]

     -1

Limit[(Sqrt[H^2+1])/(3*H-2)-1/2,H->Infinity]

       1
     -(-)
       6

Limit[(5*H+6)/(H^2+3*H+2),H->Infinity]

     0

Problem 2

Find the standard parts of each of the following numbers.

Limit[(2/(x+e)-2/x)/e,e->0]

     -2
     --
      2
     x

Limit[((H+1)^2-5)/(H^2+2*H+1),H->Infinity]

     1

Limit[(27-(3+e)^3)/e,e->0]

     -27

Problem 3

Determine whether the following functions are differentiable at the indicated point (using the definition of the derivative). If differentiable, calculate the derivative at that point.

{D[(1-x)/(1+x),x],D[(1+x)/(1-x),x]}/.{x->0}

     {-2, 2}

D[x*Abs[x^3-2],x]/.{x->0}

     2

Problem 4

Find the derivatives of the following functions:

D[x^4/2+2/5,x]

        3
     2 x

PowerExpand[D[Sqrt[x^3]-1/Sqrt[x],x]]

       1      3 Sqrt[x]
     ------ + ---------
        3/2       2
     2 x

Problem 5

Let f(x) = x^2 - 3x + 1. Find the equation of the tangent line to the graph y = f(x) at the point where x = 2.

y[x_]:=x^2-3*x+1

dy[x_]=D[y[x],x]

     -3 + 2 x

y-y[2]-dy[2]*(x-2)==0

     3 - x + y == 0

At what point on the graph y = f(x) is the tangent line parallel to the line y = 7x-6?

{x,y[x]}/.Solve[dy[x]==7,x][[1]]

     {5, 11}

alexsuciu@neu.edu

http://www.math.neu.edu/~suciu/mth1733/1733f97.ansq1.html
Created: October 2, 1997. Modified: October 20, 1997.