Graph Sketching Calculus Problems

Front Matter1 Limits and Continuity2 Differentiation Rules3 Applications of Differentiation4 Parametric Equations and also Polar Coordinates5 True or False and also Multiple Choice Problems6 Recommendations for Success in Mathematics
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1.

You watching: Graph sketching calculus problems

Give an example of a function via one critical suggest which is additionally an inflection allude. You should provide the equation of your feature.


2.

Give an instance of a role that satisfies (f(-1)=f(1)=0) and (f^prime(x)>0) for all (x) in the doprimary of (f^prime ext.)


4.

For what values of the constants (a) and also (b) does the feature (f(x)=ln a+bx^2-ln x) have actually an extremum worth (f(2)=1 ext?)


5.

The attribute (f(x)=ax^3+bx) has actually a local extreme value of 2 at (x=1 ext.) Determine whether this extremum is a local maximum or regional minimum.


6.

Prove that the attribute (f(x)=x^151+x^37+x+3) has neither a neighborhood maximum nor a neighborhood minimum.


7.

A pwrite-up moves alengthy a line via a position feature (s(t)) wbelow (s) is measured in metres and also (t) in seconds. Four graphs are displayed below: one coincides to the function (s(t) ext,) one to the velocity (v(t)) of the pshort article, one to its acceleration (a(t)) and also one is unrelated.


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Observe that the position attribute (s) is already labeled. Identify the graphs of (v(t)) and also (a(t) ext.)

Find all time intervals when the pshort article is slowing down and also when it is increasing.

Estimate the full distance took a trip by the pwrite-up over the interval (<1,6> ext.)


Velocity — bottom left; acceleration — bottom appropriate.

Speeding up on ((0,1.5)) and ((3.2,5) ext.)

Approximately 5.8 devices.


8.

Function (f) is differentiable almost everywhere. The graph of (f^prime) is illustrated in the Figure below. (f^prime) is negative and also concave dvery own at all points not displayed in this graph.

Does the feature (f) have a neighborhood maximum? If so, recognize the approximate coordinate(s) of the regional maximum point(s).

Does the attribute (f) have actually a regional minimum? If so, identify the approximate coordinate(s) of the neighborhood minimum point(s).

Does the function (f) have any inflection points? If so, determine the approximate coordinate(s) of the inflection point(s).

Determine the interval(s) on which the function (f) is decreasing.

Determine the interval(s) on which (f^primeprime) is decreasing.

If (f) is a polynomial attribute, what is the least feasible degree of (f ext?)


Answer

((3,f(3)) ext.)

((-1,f(-1)) ext.)

((-0.5,f(-0.5)) ext,) ((0.5,f(0.5)) ext,) ((2.3,f(2.3)) ext.)

((-infty,-1)) and ((3,infty) ext.)

((-infty,0)) and also ((1.5,infty) ext.)

(5 ext.) Observe that the second derivative has three zeros.


9.

Lay out the graph of (f(x)=3x^4-8x^3+10 ext,) after answering the following concerns.

Wright here is the graph increasing, and wright here is decreasing?

Wright here is the graph concave upward, and also wbelow is it concave downward?

Where are the regional minima and neighborhood maxima? Establish conclusively that they are neighborhood minima and maxima.

Wbelow are the inflection points?

What happens to (f(x)) as (x o infty) and also as (x o -infty ext.)


Answer
Solution

From (f"(x)=12x^2(x-2)) we conclude that (f"(x)>0) for (x>2) and also (f"(x)lt 0) for (xlt 2 ext.) So (f) is boosting on ((2,infty)) and decreasing on ((-infty,2) ext.)

From (f"(x)=12x(3x-4)) it complies with that (f"(x)>0) for (xlt 0) or (ds x>frac43) and also (f"(x)lt 0) for (ds xin left( 0,frac43 ight) ext.) Also (f"(x)=0) for (x=0) and also (ds x=frac43 ext.) Thus (f) is concave upward on ((-infty,0)) and also on

Critical numbers are (x=0) and also (x=2 ext.) Since (f"(x)) does not readjust sign at (x=0) tright here is no neighborhood maximum or minimum tright here. (Note additionally that (f"(0)=0) and that the second derivative test is inconclusive.) Since (f"(x)) changes from negative to positive at (x=2) there is a neighborhood minimum at (x=2 ext.) (Keep in mind likewise that (f"(2)>0 ext,) so second derivative test states tright here is a regional minimum.)

Inflection points are ((0,10)) and (ds left( frac43,fleft( frac43 ight) ight) ext.)

(ds lim _x o pm inftyf(x)= infty ext.)


10.

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In this question we consider the attribute (ds f(x)=fracx-3sqrtx^2-9 ext.)

Find the domajor of (f ext.)

Find the coordinates of all (x)- and also (y)-intercepts, if any type of.

Find all horizontal and vertical asymptotes, if any type of.

Find all important numbers, if any type of.

Find all intervals on which (f) is increasing and those on which (f) is decreasing.

Find the ((x,y)) works with of all maximum and also minimum points, if any.

Find all intervals on which (f) is concave up and also those on which (f) is concave dvery own.

Find the ((x,y)) works with of all inflection points, if any kind of.

Map out the graph of (y=f(x)) making use of every one of the above indevelopment. All relevant points need to be labeled.


Answer

From (x^2-9>0) it follows that the doprimary of the feature (f) is the set ((-infty,-3)cup(3,infty) ext.)

The attribute is not characterized at (x=0 ext,) so there is no the (y)-intercept. Keep in mind that (f(x) ot= 0) for all (x) in the doprimary of (f ext.)

From (ds lim _x o infty f(x)=1) and also (ds lim _x o -infty f(x)=-1) we conclude that there are 2 horizontal asymptotes, (y=1) (as soon as (x o infty)) and (y=-1) (once (x o -infty)). From (ds lim_x ightarrow 3^+f(x)=0) and also (ds lim_x ightarrowhead -3^-f(x)=-infty) it adheres to that tright here is a vertical asymptote at (x=-3 ext.)

Since, for all (x) in the domain of (f ext,) (displaystyle f"(x) = frac3(x-3)(x^2-9)^3/2 ot= 0) we conclude that there is no important number for the attribute (f ext.)

Note that (f"(x)>0) for (x>3) and also (f"(x)lt 0) for (xlt -3 ext.) Therefore (f) increasing on ((3,infty )) and also decreasing on ((-infty ,-3) ext.)

Since the doprimary of (f) is the union of 2 open up intervals and because the function is monotone on each of those intervals, it adheres to that the function (f) has neither (neighborhood or absolute) a maximum nor a minimum.

From (displaystyle f""(x) =-frac6(x-3)(x-frac32)(x^2-9)^5/2) it complies with that (f""(x)lt 0) for all (x) in the domajor of (f ext.) As such (f(x)) is concave downwards on its doprimary.

None.


The doprimary of the attribute (f) is the set (mathbbRackslash 0\text.) The (x)-intercepts are (pm 1 ext.) Due to the fact that 0 not in domajor of (f) tbelow is no (y)-intercept.

From (displaystyle lim_x ightarrowhead 0^- f(x) = -infty) and (ds lim_x ightarrowhead 0^+ f(x) =infty) it follows that the vertical asymptote is the line (x=0 ext.) Because (displaystyle lim_x ightarrowpminftyf(x) = lim_x ightarrowpminftyleft( x-frac1x ight)= pminfty) we conclude that tright here is no horizontal asymptote. Finally, the truth (displaystyle f(x) = x -frac1x) indicates that (f) has actually the slant (oblique) asymptote (y=x ext.)

For all (xin mathbbRackslash 0\text,) (displaystyle f"(x) = fracx^2+1x^2 >0) so the feature (f) is raising on ((-infty ,0)) and also on ((0,infty) ext.) The feature (f) has actually no critical numbers and therefore cannot have actually a local maximum or minimum.

Because (displaystyle f""(x) = -frac2x^3) it complies with that (f""(x)>0) for (xlt 0) and (f""(x)lt 0) for (x>0 ext.) As such (f) is concave upward on ((-infty ,0)) and concave downward on ((0,infty) ext.) There are no points of inflection.


12.

Given (ds f(x)=fracx^2x^2-1 ext:)

Find the domajor of (f ext.)

Is (f) an even attribute, odd function, or neither?

Find all the (x)- and also (y)- intercepts of the graph of (f ext.)

Find all horizontal, vertical, and slant asymptotes of the graph of (f ext.) If asymptote(s) of a certain type are lacking, explain why.

Find the intervals wright here the graph of (f) is increasing or decreasing, and places and also worths of the neighborhood maxima and also minima.

Find the intervals wright here the the graph of (f) is concave upward or downward and also the inflection points.

Map out the graph and also plainly indicate the features discovered in parts (a)–(f).


Answer
13.

Given (ds f(x)=fracx^2x^2+9 ext:)

Find the domajor of (f ext.)

Is (f) an even feature, odd function, or neither?

Find all the (x)- and (y)- intercepts of the graph of (f ext.)

Find all horizontal, vertical, and slant asymptotes of the graph of (f ext.) If asymptote of a details sort are lacking, explain why.

Find the intervals where the graph of (f) is boosting or decreasing, and locations and values of the local maxima and minima.

Find the intervals where the the graph of (f) is concave upward or downward and also the inflection points.