Mathematics

Problem

Consider the curve

y=kx^{r}

, defined for

x>0

, that passes through the point

(3,54)

. The line tangent to the curve at this point intersects the

y

-axis at

(0,-6)

. Parts: (a) Using only the information provided, determine the slope of the tangent line at

(3,54)

and explain your reasoning. (b) Express the derivative

\dfrac{dy}{dx}

y=kx^{r}

in terms of

k

r

, and

x

. (c) By equating your result from part (a) with the derivative evaluated at

x=3

, derive an equation that relates

k

and

r

. (d) Combine the relation from part (c) with the fact that the curve passes through

(3,54)

to solve simultaneously for

k

and

r

. Show all algebraic steps without skipping intermediate reasoning. (e) Verify that the resulting function satisfies both original geometric constraints.

Intuitive explanation

The key is to use two pieces of information—the point on the curve and the tangent line—to set up a system of equations for the unknowns $k$ and $r$ . First, substitute the given point into the curve equation to get one constraint. Then, use the fact that the tangent line passes through both $(3,54)$ and $(0,-6)$ to find the slope, which you can equate to the derivative $\frac{dy}{dx}$ evaluated at $x=3$ . This gives you a second equation to solve for both parameters.

Intuitive explanation

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