(1) Fill out the Table in Order to Find

(1) Fill out the table in order to find :

x | 1.75 1.9 1.99 1.999 2.001 2.01 2.1 2.25

f(x) |

(2) Determine the limit from the graph of the function, or state 'DNE'.

(a)

(b)

(3) Find the limit of the function using limit rules:

(a)

(b)

(4) Discuss the continuity of the function:

(a)

(b)

(5) Use the definition of derivative, , to find the derivative of .

(6) Use derivative rules to find the derivative of the function:

(a)

(b)

(c)

(d)

(e)

(f)

(7) Find by implicit differentiation:

(8) Find the equation of the line tangent to at (2,4).

(tip: find the slope here, and then use the point-slope formula for the equation of a line:

)

(9) The height of a falling object that is thrown upward with an initial velocity of 40 ft./sec., from a height of 300 ft. is given by , where t is the number of seconds since the throw. Find the velocity and acceleration of the object after 2 seconds.

(10) An alien spacecraft is touching down in a cornfield in Bamberg. As it approaches, the wind from the landing mechanism creates a crop circle, expanding at the rate of 2 ft./second. How fast is the area of the crop circle increasing when it's radius is 6 ft?

(11) A company's profit from selling x units of it's product can be modeled by

P = 700x - . The sales are increasing at a rate of 8 per day. Find the rate of change of profit when 500 units have been sold.

Solution:

(1) x | 1.75 1.9 1.99 1.999 2.001 2.01 2.1 2.25

f(x) | 3.75 3.9 3.99 3.9999 4.001 4.01 4.1 4.25

f(x) is approaching 4 from both sides, so = 4

(2) (a) = 2, as the function is 'zooming in' on this value from both sides of x = 0.

(b)

(3) (a) just plug in x = 1,

= (1)3 - 4(1)2 +3(1) + 2 = 2.

(b) If you try to plug in x = 3 here, you get 0/0. But you can cancel out the problem factor, and then plug in x = 3:

=== .

(4) (a) This function is a polynomial, and polynomials are continuous for every value of x on the real line .

(b) This function is continuous except where the denominator is equal to zero, set

So, f is continuous over - {-4, 4}

(5)

(6) (a)

(b) You need to rewrite the terms before you use the power rule:

f '(x) = (2)(x2+1)-(2x -3)(2x) = 2x2+2 - 4x2 - 6x = -2x2 - 6x + 2

(x2+1)2 (x2+1)2 (x2+1)2

(d) Because we're multiplying the functions, we have to use the product rule here:

g'(x) = (2x - 2)(x4 - 4x2 + 7x) + (x2 - 2x + 3)(4x3 - 8x + 7)

(e) Here, we have an 'inside' function and an 'outside' function - we need to use the chain rule.

(diff the outside first, then chain on the derivative

h'(x) = 5(x2- 3x + 7)4 (2x - 3)

(f) First, rewrite the square root as a power, then use the chain rule again:

g(x) = (x2 - 3x)½

g'(x) = ½(x2 - 3x)-½ = 1___

2√ x2- 3x

(7)

(8) Slope is the same as derivative, so you need to find :

then plug in the x-value x = 2 (the point is (2,4)),

so m = 4,

Then use the point-slope formula:

(9) You should know that we associate velocity with the first derivative, and acceleration with the second derivative.

(10) First, the givens translate into: when r = 6.

The fundamental relationships between the variables arises from the formula for the area of a circle:

(now, implicitly differentiate to introduce the derivatives:)

(plug in the givens to find dA: )

This is the rate of expansion of the area when r = 6.

(11) First, translate the info about the givens - sales increasing 8 per day means:

and we want when x = 500. First the fundamental relationship is:

P = 700x - (differentiate implicitly:)

(plug in the givens: )

Whatever that is.