Find a general formula for fx fxh-fxh as hrarr0 steps 1

Question

I. Computing Derivatives (slope of curve at a point) of polynomial functions.

For each of the following functions in a.-e. below perform the following three steps:

1. Compute the difference quotient f(x)=[f(x+h)-f(x)]/h

2. Simplify expression from part 1. such that h has been canceled from the denominator

3. Substitute h=0 and simplify

a. f(x)=c

b. f(x)=ax+b

c. f(x)=ax²+bx+c

d. f(x)=ax³+bx²+cx+d

e. consider f(x)=a_nxⁿ+a_n-1x^n-1+....a₁x+a₀, using the results from parts a. through d.,

f. find a general formula for f(x)= [f(x+h)-f(x)]/h as h→0 (steps 1 through 3 performed).

II. Show that sinθ/θ=1 as θ→0

Consider the unit circlewith θ in standard position in QI.

a. set up the inequality [(tanθ)/2]≥θ/2≥[(sinθ)/2]

b. multiply the inequality in part a. by 2/sinθ. (direction of inequalities isunchanged)

c. take the reciprocal of each term from part b. The direction of the inequality must be reversed because if a,b>0→a

d. plug in 0 for θ for cosθ only. The result should be 1≤(sinθ)/θ≤1 as θ→0

III. Show that (1-cosθ)/θ=0 as θ→0

a. multiply (1-cosθ)/θ by (1+cosθ)/(1+cosθ)

b. use trigonometric identity cos²θ+sin²θ=1 to rewrite the numerator of the expression in part a. in terms of sin²θ

c. factor the expression in part b. with one factor equal to (sinθ)/θ. (find remaining factor).

d. use the fact that (sinθ)/θ=1 as θ→0 and substitute θ=1 in the second factor (result is 0)

IV. Show that derivative of sinθ=cosθ

a. find the difference quotient f(θ)=[f(θ+h)-f(θ)]/h for f(θ)= sinθ

[use sum angle formula sin(θ+h)=sinθcosh+cosnθsinh]

b. factor sinθ out of the two terms in the numerator with sinθ in part a

c. split up the expression in part b with each term over the denominator h

d. use identities (sinh)/h=1 and (1-cosh)/h=0 as h→0 to simplify part c. to cosθ

Thus you have shown that if f(θ)=sinθ, then f(θ)=[f(θ+h)-f(θ)]/has h→0=cosθ.

Answer 1

Solve each and every part of the mathematics question (Computing Derivatives of polynomial functions) which is illustrated below by using the appropriate concepts and theories. In this, you have to perform the following three steps: 1) Calculate the difference quotient f(x) = [f(x+h)-f(x)]/h 2) Solve the expression from part 1 in such a way that ‘h’ has been canceled from the denominator 3) Replace h = 0 and simplify a) f(x) = c b) f(x) = ax + b c) f(x) = ax2 + bx + c d) f(x) = ax3 + bx2 + cx + d e) Consider f(x) = anxn + an-1xn - 1+....a1x+a0, by employing the results from parts a. via d. f) Determine a general formula for f(x) = [f(x+h) - f(x)]/h as h?0 (steps 1 via 3 performed).