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equation of the tangent line to the graph

1) Let

f(x)= {-x+b, if x<-1
{5, if x=-1
{(-5/(x-b))+4, if x>-1 (and x=b)

a) For what value(s) of b in f continuous at -1?
b=________

b) For what value(s) of b does f have a removable discontinuity at -1?
b=________

c) For what value(s) of b does f have an infinity discontinuity at -1?
b=________

d) For what value(s) of b does f have a (finite) jump discontinuity at &#8722;1? Write your answer in interval notation.
b is in the set=__________

2. Suppose that A is a constant and f(x) is a function of x such that

((Ax)/(x-2)) < f(x) < x-512 for all x near 32 but not equal to 32. We are interested in finding the limit of f(x) as x approaches 32 by means of the Squeeze Theorem.

a) For the Squeeze Theorem to be applicable in this case, the constant A must be equal to a specific number. Find this number.
A=________

b) Assuming that A is that number for which the Squeeze Theorem is applicable, find lim f(x)
x->32
This limity is equal to=_________

3.Evaluate the following limits, assuming that all angles are in radian.

a) lim (sin5x)/(sin4x) =________
x->0

b)lim (xsin3x)/(sin^2 9x) =___________
x->0

c) lim (sin4x)/(9x-5tanx) =__________
x->0

4. Consider the function f(x)= 2/(x-6).
We will take steps to find the tangent line to the graph of f at the point (3,2/-3)

a) Let (xf(x)) be a point on the graph of f with x=3 . The slope of the (secant) line joining the two points (3,2/-3) and (x,f(x)) can be simplified to the form A/x-6, where A is a constant. Find A.
A=_________

b) By considering the slope of the secant line as x approaches 3, find the slope of the tangent line to the graph of f at the point (3,2/-3)
The slope of the tangent line to the graph of f at the point (3,2/-3) is =__________

c) Find the equation of the tangent line to the graph of f at the point (3,2/-3). Write your answer in the form y=mx+b.
The equation of the tangent line to the graph of f at the point (3,2/-3) is y=__________

5. Consider a moving object whose displacement at time t is given by s(t)= -4t^2-7t.
We will take steps to find the instantaneous velocity of the object at time t=8.

a) For any time t=8 the average velocity of the object on the time interval between 8 and t can be simplified into the form At+B, where A and B are constants. Find these constants.
A=_________, B=__________

b) By considering the average velocity on the shrinking time interval between 8 and t as t approaches 8, determine the instantaneous velocity of the object at time 8.
The instantaneous velocity of the object at time 8 is =________

Solution Preview

1) Let

f(x)= {-x+b, if x<-1
{5, if x=-1
{(-5/(x-b))+4, if x>-1 (and x=b)

a) For what value(s) of b in f continuous at -1?
b=4
f approaches 1+b from left side of -1 and f approaches -5/(-1-b) + 4 from the right side of -1.
We set 1+b = -5/(-1-b) + 4 and solve the equation. Then we get b = -2 or b = 4
Now since f(-1) = 5 and when b=4, 1+b = -5/(-1-b) + 4 = 5 = f(-1), so the answer is b=4

b) For what value(s) of b does f have a removable discontinuity at -1?
b=-2
From a)'s solution, we know another choice of b is -2 and in this case, 1+b = -5/(-1-b) + 4 = -1, not equal to f(-1).
So f has removable discontinuity at -1.

c) For what value(s) of b does f have an infinity discontinuity at -1?
b=-1
To find this, we only need to set the denominator x-b = -1-b = 0, then b=-1

d) For what value(s) of b does f have a (finite) jump discontinuity at &#8722;1? Write your answer in interval notation.
b is in the set=(-oo, -1) U (-1, -2) U (-2, 4) U (4, oo)
Excluding the ...

Solution Summary

Find the equation of the tangent line to the graph.

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