If $u$ can do it, then $du$ can undo it!

Undoing the Chain Rule

7-53.

(NOT SO) HARD-LOOKING INTEGRALS, Part One

Work with your team to evaluate these integrals. Some look intimidating, but all have a similar structure. Check your answers as you go—you will not see the patterns if you are not $100\%$ confident your answers are correct.

$\int\sin(x^2) · (2x)dx$

$\int \sin(x) · e^{\cos(x)}dx$

$\int (7x^2 + 1)^4 · (14x)dx$

$\int_{ }^{ }(1+\cos(x))^4·\sin(x)dx$

$\int \sin(5x) · \cos^2(5x)dx$

$\int (x^2 + 2x + 5)^{7/3} · (x + 1)dx$

How did you decide what constant(s) you needed to multiply by? Write a complete explanation.
In each case you had to undo a complicated expression. Describe your process and include the following words in your explanation: composite function, derivative, outer function, and inner function.
Use your pattern to write another, similar problem. Your new integrals should be challenging, and each team member should write his/her own.

7-54.

Cone with vertex on the bottom, above a cylinder, with bottom half of cone shaded, vertical distance of the shaded portion labeled, h, vertical distance of the cone labeled, 6 inches, radius of top base labeled, 3 inches, cylinder with bottom fourth shaded, vertical distance of the shaded portion labeled, y, distance across circular bottom labeled, 6 inches. Water drips from a conical filter into a coffee mug as shown at right. Suppose $h$ is the height of the water in the filter, $y$ is the height of the water in the cup, and there is a total of $9$ in³ of water in the filter and cup. Write an equation relating $\frac { d y } { d t }$ and $\frac { d h } { d t }$ .

7-55.

Differentiate each equation. Homework Help ✎

$y^2 = 3\cos(x)$

$y = \tan^2( \sqrt { x } )$

$y = \sin(\cos(x))$

7-56.

Integrate. Homework Help ✎

$\int(3x − 5)^2dx$

$\int _ { - 3 } ^ { 3 } 9 \operatorname { sin } ( \pi x ) d x$

$\int(6m^{−7} + 2)dx$
$m$ is a constant

7-57.

Sam the snowman has a spherical head that is melting at a rate of $12 \text{ in}^3$ per hour. Assume that as it melts, his head remains spherical. What is the radius of Sam’s head when the radius is changing at $0.25$ inches per hour? Homework Help ✎

7-58.

What is the value of $a$ such that the parabola $y = ax^2 + 3$ is tangent to the line $y = 5 - x$ ? Also, state the point of tangency. Homework Help ✎

7-59.

The value of a new car is $$10,000$ . The value decreases by $2\%$ each year.
What is the car’s average value during the first $12$ years? Homework Help ✎

7-60.

The velocity of a plane flying from San Francisco is given by $v(t) = 270\sqrt { t }$ where $t$ is measured in hours and $v(t)$ in miles per hour. Calculate the average velocity of the plane between the first and fourth hours. Homework Help ✎

7-61.

No calculator! Given $f(x) = 3^x – 2$ and $g(x) =\frac { 1 } { 2 }\sin(x)$ , evaluate each of the following limits. Homework Help ✎

Compute without a calculator

$\lim\limits _ { x \rightarrow 1 }f ^\prime(x)$

$\lim\limits _ { x \rightarrow \pi / 4 } g ^ { \prime \prime } ( x )$

$\lim\limits _ { x \rightarrow \pi }f (g(x))$

$\lim\limits _ { x \rightarrow \pi / 2 } f ( g ^ { \prime } ( x ) )$

$\lim\limits _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$

$\lim\limits _ { x \rightarrow \pi / 4 } \frac { g ( x ) - \frac { \sqrt { 2 } } { 4 } } { x - \frac { \pi } { 4 } }$

Calculus Third Edition