Find the function The following limits represent the slope of a curve y=f(x) at the point {a,f(a)} Determine a possible function f and number a, then calculate the limitlim_{xrightarrow2}frac{5cdot x^2-20}{x-2}

EunoR 2021-02-26 Answered

Find the function The following limits represent the slope of a curve \(y=f(x)\) at the point \(\{a,f(a)\}\) Determine a possible function f and number a, then calculate the limit
\(\lim_{x\rightarrow2}\frac{5\cdot x^2-20}{x-2}\)

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Expert Answer

StrycharzT
Answered 2021-02-27 Author has 12588 answers

Consider the given limit \(\lim_{x\rightarrow2}\frac{5\cdot x^2-20}{x-2}\)
Note that, \(f'(a)=\lim_{x\rightarrow a}\frac{f(x)-f(a)}{x-a}\)
That implies, \(f(x)=5x^2\) and \(a=2\)
Now obtain the limit as follows
\(\lim_{x\rightarrow2}\frac{5\cdot x^2-20}{x-2}=\lim_{x\rightarrow2}\frac{5(x^2-4)}{x-2}\)
\(=\lim_{x\rightarrow2}\frac{5(x^2-2^2)}{x-2}\)
\(=\lim_{x\rightarrow2}\frac{5(x-2)(x+2)}{(x-2)}\)
\(=\lim_{x\rightarrow2}5(x+2)\)
\(=5(2+2)\)
\(=5\cdot4\)
\(=20\)
Hence, the value of the given limit is 20.

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Consider the curves in the first quadrant that have equationsy=Aexp(7x), where A is a positive constant. Different valuesof A give different curves. The curves form a family,F. Let P=(6,6). Let C be the number of the family Fthat goes through P.
A. Let y=f(x) be the equation of C. Find f(x).
B. Find the slope at P of the tangent to C.
C. A curve D is a perpendicular to C at P. What is the slope of thetangent to D at the point P?
D. Give a formula g(y) for the slope at (x,y) of the member of Fthat goes through (x,y). The formula should not involve A orx.
E. A curve which at each of its points is perpendicular to themember of the family F that goes through that point is called anorthogonal trajectory of F. Each orthogonal trajectory to Fsatisfies the differential equation dy/dx = -1/g(y), where g(y) isthe answer to part D.
Find a function of h(y) such that x=h(y) is the equation of theorthogonal trajectory to F that passes through the point P.
...