Calculus Homework Question? 5 Star best answer?

The straight line y = 1 + 3x has a gradient of 3.

The tangent will also have a gradient of 3, since it is parallel to this line.

Thus its equation will be y = 3x + c.

Also (3/2)x^(1/2) = 3, from which you can find a value for x (I make it 4).

Thus y = x√x = 4√4 = ±8

The tangent meets the curve at (4, 8) or (4, -8)

The equation of the tangent is y = 3x + c

For (4, 8)

8 = 3(4) + c

8 = 12 + c

c = -4

The equation of the tangent is y = 3x - 4

For (4, -8)

-8 = 3(4) + c

-8 = 12 + c

c = -20

The equation of the tangent is y = 3x - 20

Let f(x) = x^3/2

We have f defined for x being greater than or equal to 0.

The equation of the tangent to (Cf) at any point M(a,b) is:

(T): y - b = f '(x) (x - a) <=> y = f '(x) . x - af '(x) + b

As (T) is parallel to y = 3x + 1; then, slope(T) = 3 => f '(x) = 3

(Two lines are parallel if and only if they have the same slope)

f '(x) = (3/2)x^(1/2)

f '(x) = 3 <=> (3/2)x^(1/2) = 3 <=> x^(1/2) = 2 <=> x = 4

So x = 4 = a

f(4) = 8 = b

Thus we have a and b and f '(x), replace them in the equation of (T) to get your desired result.

d/dx √x = d/dx x^(1/2) = (1/2) x^(-1/2) = 1 / 2x^(1/2) = 1 / 2√x

f(x) = x √x

f'(x) = √x + x [ 1/ 2√x]

f'(x) = √x +√x/2 = 3√x/2

The parallel line has the same slope

slope of the parallel line = 3

3√x/2 = 3

√x/2 = 1

√x 2

x=4

y = x√x = (4) √4 = 8

Equation of the tangent line at (4,8):

y-8 = 3 ( x-4)

y = 3x - 4

y = 1 + 3x tells you that slope is 3

y' = 3 = (3/2) x^(1/2)

x^(1/2) = 2 .... x = 4

y = x^(3/2) = 8 .... so slope 3 thru (4,8)

y-8 = 3(x-4)

y = 3x +4