The Derivative

Question

Compute The Derivative of the given function and find the equation of the tangent line that is tangent to it's graph for the specified value x=c

14.) f (x) = 3; c= -4
16.) f(x) = 3x; c=1

 

find the rate of change dy/dx where x= X°

30. y=x^2 - 2x;       X°=1

38 Let f(x) = x(1-2x).

a. find the ave. rate of change of f(x) with respect to x as x change from x=0 to x=1/2

b. uce calculus to find the instantaneous rate of change of f(x) at x = 0 and compare with the ave. rate found in part A.

 

 

If someone could help me it would be the greatest thing for today.

Answer 1

14)

 

The derivative a constant function is always zero.  This is the slope of the tangent line.   The tangent line passes through the point (3, -4).

 

Plugging in the slope and coordinate values into the slope-intercept form of a line to find b,

 

y = mx + b

 

-4 = 0(x) + b

 

-4 = b

 

 

So the equation of the line we get is      f(x) = -4.  This line is not tangent to the given line.  Rather, it is parallel to the given line.

 

 

 

15)

 

Do the same thing as in the last question.  The tangent line passes through the point (1, 3).  The derivative of f(x) is 3.  This is the slope of the tangent line.  Plugging in this slope and coordinate values into the slope-intercept form of a line to find b,

 

y = mx + b

 

3 = 3(1) + b

 

3 = 3 + b

 

0 = b

 

 

So the equation of the tangent line is then     f(x) = 3x, which is the same line we started with.

 

 

30)

 

Find the derivative of f(x).  Then plug in x=1.

 

f'(x) = 2x - 2

 

Evaluate f'(1).

 

 

38.  Rewrite the function so that it is easier to work with.

 

f(x) = x - 2x²

 

 

Now do parts a and b.

 

38a)

 

Use the formula

 

average rate of change = [f(b) - f(a)] / (b - a)

 

where:

a = 0

b = 1/2

 

 

38b)

 

Use the definition of limit.  This is the same as the derivative.

 

lim          f(x + h) - f(x)

h-->0     _____________

                       h