Exam 1

MAT 128, Fall 2016

Show your work to receive credit

Name:

Use the graph of g(x) below to answer the following questions. (4 points each)

f1[x_]=((1/2)/(Abs[x+3]))-2;
f2[x_]=(x+2)-3/2;
f3[x_]=4-(x^(1/2));
f4[x_]=(((x-4))^2)+2;
Show[Plot[-f1[x]+4,{x,-6,-2},PlotRange->All],Plot[-f2[x]+4,{x,-2,0.8}],Plot[-f3[x]+4,{x,1.2,3.8}],Plot[-f4[x]+4,{x,4.2,6}],Graphics[{Circle[{4,-2+4},0.2],Circle[{1,-f2[1]+4},0.2],Disk[{1,-f3[1]+4},0.22]}], PlotRange->{{-6,6},{-6,6}},AspectRatio->Automatic,AxesOrigin->{0,0},GridLines->{Range[-6,6],Range[-6,6]},GridLinesStyle->{GrayLevel[0.5],Thickness[0.003]},AxesStyle->Thickness[0.007],PlotLabel->HoldForm[y=g[x]],AxesLabel->{x,y}]

Based on this graph, are there any values in [-5,5] not in the domain of g(x)? If so, which values?

Does (lim_(x->-3))g(x) exist or not? If it exists, what is its value?

Does (lim_(x->-2))g(x) exist or not? If it exists, what is its value?

Does (lim_(x->1))g(x) exist or not? If it exists, what is its value?

Does (lim_(x->4))g(x) exist or not? If it exists, what is its value?

The temperature of a meal in a pressure cooker is provided in the following table:
	({{t (minutes), 36, 38, 40, 42, 44}, {temp((^o)F), 153, 166, 180, 194, 207}}) 	(3 points each)

What is the average rate of change of temperature (per minute) from t=36 to t=42?

What is the average rate of change of temperature (per minute) from t=38 to t=42?

What is the average rate of change of temperature (per minute) from t=40 to t=42?

Produce a reasonable estimate for the instantaneous rate of change of temperature (per minute) at t=42, with justification.

Use the following table of values for f(x) to decide if you believe (lim_(x->2))f(x) exists or not. If you say it does exist, estimate its value. If you say it doesn’t exist, explain what makes you think that. (6 points)

({{x, f(x), x, f(x)}, {1.0, 2.000000, 3.0, 9.000000}, {1.5, 2.750000, 2.5, 6.750000}, {1.8, 3.440000, 2.2, 5.640000}, {1.9, 3.710000, 2.1, 5.310000}, {1.95, 3.852500, 2.05, 5.152400}, {1.99, 3.970100, 2.01, 5.030100}, {1.995, 3.985025, 2.005, 5.015025}, {1.999, 3.997001, 2.001, 5.003001}})

Determine the exact value of each limit, if it exists. Show your work or otherwise justify your answers. (8 points each)

(lim_(x->-3))(((x^2)-9)/((x^2)+2x-3))

(lim_(x->0))(((Sqrt[5+x])-(Sqrt[5]))/x)

(lim_(x->2))(((x^3)+8)/((x^2)+4x+4)) (describe all steps)

(lim_(x->-1))(((x^2)-3x+2)/((x^2)+2x+1))

(lim_(x->1))h(x) where h(x)={({{3x+5, if, x<-1}, {2(x^2), if, -1≤x<1}, {2cos(x-1), if, x>1}})

Let g(x)=(((x^2)-2)/(x+1)).  Here is the graph of g in the vicinity of the origin:

Find an equation in point-slope form for the secant line s(x) to y=g(x) using x-values -2 and -3/2. Draw the secant line into the graph above, and use your equation to estimate the location of a root of g(x). (11 points)

Suppose that your first guess for the root of g(x) is x=0. Draw the tangent line on the graph above, estimate its slope (show your work), and use your tangent line to produce an improved guess for the root of g(x). (11 points)

g[x_]:=(x^2-2)/(x+1)
Plot[g[x],{x,-2,2}]
Plot[{g[x],-2+3(x+2)},{x,-2,2}]
g'[0]
Plot[{g[x],-2+3(x+2),-2+g'[0]x},{x,-2,2}]