Exam 3: Applications of Differentiation

Two curves are said to be orthogonal if their tangent lines are perpendicular at each point of intersection of the curves. Show that the curves of the given equations are orthogonal. y - $\frac{3}{4}$ x = $\frac{\pi}{2}$ x = $\frac{3}{4}$ cos y

Find the derivative of the function. $\sqrt[8]{x}+\frac{1}{\sqrt[6]{x}}$

Find the derivative of the function. $y=(3 x+1)^{9}\left(x^{4}-6\right)^{3}$

A spherical balloon is being inflated. Find the rate of increase of the surface area $S=4 \pi r^{2}$ with respect to the radius r when r = 3 ft.

Use implicit differentiation to find dy/dx. $\ln x y-y^{3}=9$

Calculate $y^{\prime }$ . $x y^{5}+x^{4} y=x+3 y$

Find the given integral. $\int \cosh (9 x+5) d x$

Find dy/dx by implicit differentiation. $e^{x y}-x^{8}+y^{8}=8$

The mass of part of a wire is $x(1+\sqrt{x})$ kilograms, where x is measured in meters from one end of the wire. Find the linear density of the wire when x = 16 m .

Find the derivative of the function. y = $\sqrt{36 x^{2}-1}-6 \cosh ^{-1} 6 x$

(a) Find the number c whose existence is guaranteed by the Mean Value Theorem for Integrals for the function f on [a, b], and (b) sketch the graph of f on [a, b] and the rectangle with base on [a, b] that has the same area as that of the region under the graph of f. $f(x)$ = $\frac{1}{2}$ $x^{2}$ + x; [0, 1]

Differentiate. $y=\frac{\tan x-4}{\sec x}$

Find the tangent line to the ellipse $\frac{x^{2}}{24}+\frac{y^{2}}{6}=1$ at the point $(2,-\sqrt{3})$ .

Use logarithmic differentiation to find the derivative of the function. $y=x^{6 x}$

Find the value of the expression accurate to four decimal places. sinh 4

Find $\frac{d^{5}}{d x^{5}}\left(x^{4} \ln x\right)$ .

Gravel is being dumped from a conveyor belt at a rate of 32 ft/min and its coarseness is such that it forms a pile in the shape of a cone whose base diameter and height are always equal. How fast is the height of the pile increasing when the pile is 10 ft high? Round the result to the nearest hundredth.

The height (in meters) of a projectile shot vertically upward from a point $5.5$ m above ground level with an initial velocity of 25.48 m/s is $h=5.5+25.48 t-4.9 t^{2}$ after t seconds. a. When does the projectile reach its maximum height? b. What is the maximum height?

The circumference of a sphere was measured to be 88 cm with a possible error of $0.7$ cm. Use differentials to estimate the maximum error in the calculated volume.

Evaluate $\lim _{x \rightarrow \infty} \frac{\sinh x}{8 e^{x}}$