# Lead

### Math help?

Suppose f(x)=8x^2+5x−6. Evaluate the limit by using algebra to simplify the difference quotient (in first answer box) and then evaluating the limit (in the second answer box).

limh→0(f(−2+h)−f(−2)/h)=limh→0( )= -1/3

Appreciated

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1 AnswerMathematics2 weeks ago### Math help?

Suppose f(x)=3/x−2. Evaluate the limit by using algebra to simplify the difference quotient (in first answer box) and then evaluating the limit (in the second answer box).

limh→0(f(5+h)−f(5)/h)=limh→0(

)=

.

2 AnswersMathematics2 weeks ago### Math help. what is my mistake?

comparing secant slope through A:(xA,yA)A:(xA,yA) and B:(xB,yB)B:(xB,yB)

taking AA and BB as points “on” the function y=f(x)y=f(x), we have yA=f(xA)yA=f(xA) and yB=f(xB)yB=f(xB)

so A:(xA,f(xA))A:(xA,f(xA)) and B:(xB,f(xB))B:(xB,f(xB))

secant slope: ΔyΔx=f(xB)−f(xA)xB−xAΔyΔx=f(xB)−f(xA)xB−xA

instantaneous (or “tangent”) slope requires AA and BB to be “infinitely close”

fix the point where we want instantaneous slope: A:(a,f(a))A:(a,f(a))

let point BB vary, so that it can “approach” AA: B:(x,f(x))B:(x,f(x))

then secant slope: ΔyΔx=f(x)−f(a)x−aΔyΔx=f(x)−f(a)x−a

note that ΔyΔxΔyΔx is undefined if we let x=ax=a

in other words, we cannot make BB and AA be the same point... of course, slope needs two points!

Compute secant slopes of f(x)f(x) through AA

f(x)=36−x2−−−−−−√f(x)=36−x2

approximate the instantaneous slope at: A:(−3,27−−√)A:(−3,27)

secant slope from the left: secant slope from the right:

xx f(x)f(x) ΔyΔxΔyΔx xx f(x)f(x) ΔyΔxΔyΔx

−3.5−3.5

4.8734

0.6455

−2.5−2.5

5.4544

0.458349

−3.05−3.05

5.16696

0.5838

−2.95−2.95

5.2247

0.57095

−3.005−3.005

5.1933

0.577992

−2.995−2.995

5.1990

0.576068

Notes:

Your answers must be given as decimal approximations.

When completing the table, make sure your answers are accurate to at least 4 decimal places.

Be careful when computing the slope, you may need more than 4 decimal places for the yy-coordinate in order for the slope to achieve the required accuracy.

Estimate the instantaneous slope of f(x)f(x) at AA

Usi

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