Exam 2

MAT 128, Fall 2016

Show your work to receive credit

Name:

Match the functions (a-d) with the derivatives (I-IV), writing the corresponding letter or number in the title of each graph. GIVE REASONS for your choices (and “by process of elimination” will not be counted as a legitimate reason).  (10 pts)

f1[x_]:=Sin[x];
f2[x_]:=x^2-3x+2;
f3[x_]:=Tan[x];
f4[x_]:=Sqrt[x];
p1=Plot[f1[x],{x,0,Pi},PlotLabel->"a ___"];
p2=Plot[f2[x],{x,0,Pi},PlotLabel->"b ___"];
p3=Plot[f3[x],{x,0,Pi},PlotLabel->"c ___"];
p4=Plot[f4[x],{x,0,Pi},PlotLabel->"d ___"];
p5=Plot[f2'[x],{x,0,Pi},PlotLabel->"I ___"];
p6=Plot[f3'[x],{x,0,Pi},PlotLabel->"II ___"];
p7=Plot[f4'[x],{x,0,Pi},PlotLabel->"III ___"];
p8=Plot[f1'[x],{x,0,Pi},PlotLabel->"IV ___"];
GraphicsGrid[{{p1,p2},{p3,p4},{p5,p6},{p7,p8}}]

A rechargeable battery is plugged into a charger. The graph shows C(t), the percentage of full capacity that the battery reaches as a function of time t elapsed (in hours).  (10 pts)

Carefully sketch the graph of  C’(t) below. Roughly appropriate numerical values are important!

Plot[Sin[x]*I,{x,0,2Pi},PlotRange->{{0,12},{-60,60}}]

What is the actual meaning of the derivative C’(t)? Explain what it tells us about the battery charging process.

Use the most important definition in calculus -- the limit definition of the derivative -- to find the derivative of f(x)=(x^2)+2x-3 at x=2, and write the equation of the tangent line to the curve there. (10 pts)

Continuity (10 pts)

Describe accurately the relationship between continuity and differentiability. (Figures are encouraged.)

Give examples of the kinds of discontinuities that rational functions (ratios of polynomials) can have. Both graphs and example expressions are encouraged for full credit.

Find the derivatives of the following functions, using any means you like (but show any work). (10 pts)

f(x)=(x^2)+2x-3

f(x)=tan(x)((x^2)+2x-3)

f(x)=(x/((x^2)+x))

f(x)=|x|

Trig functions  (10 pts)

Draw careful graphs of sine and cosine that illustrate their relationships to each other and to their derivatives. I need you to then explain these relationship IN WORDS.  (5 pts)

Plot[Sin[x]*I,{x,0,2Pi},PlotRange->{{0,2Pi},{-2,2}}]

Find the equation of the tangent line to the graph of sine at the origin.  (3 pts)

Use your result from 6.2 to justify that (lim_(x->0))((sin(x))/x)=1. (2 pts)