Math Homework Help!

1. Let f (x) = x2, −3 ≤ x ≤ 3. Let A = 4. Use the the �−δ definition of limit to show that

lim x→2

f (x) = 4

You are required to express δ in terms of �. Then use such an expression to obtain the following:

[a] For a given � = 0.5, find the corresponding δ. Then obtain the domain of x around 2, given δ. And finally the range of f (x) around 4, given �.

[b] For a given � = 0.1, find the corresponding δ. Then obtain the domain of x around 2, given δ. And finally the range of f (x) around 4, given �.

[c] For a given � = 0.05, find the corresponding δ. Then obtain the domain of x around 2, given δ. And finally the range of f (x) around 4, given �.

[d] For a given � = 0.01, find the corresponding δ. Then obtain the domain of x around 2, given δ. And finally the range of f (x) around 4, given �.

[e] Plot the intervals around 2 on the x-axis and the intervals around 4 on the y-axis as obtained in [a]-[d]. Comment on your results.

Differentiation- interpreted as Rate of Change

2. Given a function f (x) = x3. You are required to find the rate of change of f (x) at a given point x = 2 as follows:

[a] Set ∆x = 1, find ∆f (x), which is defined as f (x + ∆x) − f (x) at x = 2. Then find the rate of change accordingly.

[b] Set ∆x = 0.5, find ∆f (x), which is defined as f (x + ∆x) − f (x) at x = 2. Then find the rate of change accordingly.

[c] Set ∆x = 0.1, find ∆f (x), which is defined as f (x + ∆x) − f (x) at x = 2. Then find the rate of change accordingly.

[d] Set ∆x = 0.01, find ∆f (x), which is defined as f (x + ∆x) − f (x) at x = 2. Then find the rate of change accordingly.

[e] Tabulate ∆x, ∆f (x) and rate of change of f (x) at x = 2 obtained in [a] to [d]. Comment on your results.

3. [a] Write down an expression for (a + b)n, where a and b are real numbers and n is a

positive interger.

[b] Now set a = x and b = ∆x. Obtain an expression for (x + ∆x)n.

[c] Now obtain an expression for (x + ∆x)n − xn

[d] Use the result in [c] to get the derivative of f (x) = xn with respect to x. That is to find

f′(x) = lim ∆x→0

f (x + ∆x) − f (x) ∆x

[e] Find f′(2) for n = 3. Compare your result with that obtained in [2d]. Is there is a difference? Why?

Applications 4. Plot the following functions and explain in details whether they are continuous or discon- tinuous at x = 0 for [a]-[b], and at x = 1 for [c]. Also comment on the existence of limits when x tends to the designated points.

[a]

f (x) =

 

x2, for x > 0 2, for x = 0 −x2, for x > 0

[b]

f (x) =

 

x2, for x < 0 1, for x > 0 2, for x = 0

[c]

f (x) = { 1

1−x , for x < 1 x, for x ≥ 1

5. [a] You are given a function f (x). Suppose that it is differentiable at point (a, f (a)), with

slope f′(a) at x = a. Now let the equation of the tangent to the graph of the given function f (x) at the point (a, f (a)) be y = mx + c, where m is the slope and c is the vertical intercept. Based on the given information, show that the equation of this tangent line is: y = f′(a)(x − a) + f (a).

[b] Find the derivative of f (x) = x 1 4 . Use the result in [a] to approximate the value of

f (1.08). Compare your result with that obtained by using a pocket calculator. What’s the error?

[c] Assume that a given function f (x) is twice-differentiable. That is you can differentiate this function with respect to x two times and f′′(x) exists. However we don’t know the functional form of f (x). We are going to approximate f (x) by the following quadratic polynomial

g(x) = P + Q(x − a) + R(x − a)2

where a is a fixed number, P, Q and R are coefficients to be determined. First, f (x) and g(x) should have the same value for a given x. Second, f (x) and g(x) should have the same first derivative. And third, f (x) and g(x) should have the same second derivative. Based on such arguments, show that the quadratic approximation to f (x) about x = a is:

f (x) ≈ f (a) + f′(a)(x − a) + 1 2 f′′(a)(x − a)2, x close to a

[d] Find the quadratic approximation to f (x) = x 1 4 about x = 1. Hence compute the

approximate of f (1.08). Compare your result with that obtained by using a pocket calculator. What’s the error? Also, compare your result with that obtained in [b].