Throwing a stone Suppose a stone is thrown vertically upward from the edge of a cliff on Earth with an initial velocity of 32 ft/s from a height of 48 ft above the ground. The height (in feet) of the stone above the ground t seconds after it is thrown is s(t) = -16t²+32t+48.

c. What is the height of the stone at the highest point?

Suppose a stone is thrown vertically upward from the edge of a cliff on Earth with an initial velocity of 19.6 m/s from a height of 24.5 m above the ground. The height (in meters) of the stone above the ground t seconds after it is thrown is s(t) = -4.9t²+19.6t+24.5.

c. What is the height of the stone at the highest point?

Suppose a stone is thrown vertically upward from the edge of a cliff on Earth with an initial velocity of 64 ft/s from a height of 32 ft above the ground. The height (in feet) of the stone above the ground t seconds after it is thrown is s(t) = -16t²+64t+32.

c. What is the height of the stone at the highest point?

Initial velocity Suppose a baseball is thrown vertically upward from the ground with an initial velocity of v0ft/s Its height above the ground after t seconds is given by s(t) = -16t²+v0t. Determine the initial velocity of the ball if it reaches a high point of 128 ft.

Comparing velocities Two stones are thrown vertically upward, each with an initial velocity of 48 ft/s at time t=0. One stone is thrown from the edge of a bridge that is 32 feet above the ground, and the other stone is thrown from ground level. The height above the ground of the stone thrown from the bridge after t seconds is f(t) = − 16t²+48t+32. and the height of the stone thrown from the ground after t seconds is g(t) = −16t²+48t.

a. Show that the stones reach their high points at the same time.

Matching heights A stone is thrown with an initial velocity of 32 ft/s from the edge of a bridge that is 48 ft above the ground. The height of this stone above the ground t seconds after it is thrown is f(t) = −16t²+32t+48 . If a second stone is thrown from the ground, then its height above the ground after t seconds is given by g(t) = −16t²+v0t, where v0 is the initial velocity of the second stone. Determine the value of v0 such that both stones reach the same high point.

Velocity of a car The graph shows the position s=f(t) of a car t hours after 5:00 P.M. relative to its starting point s=0,where s is measured in miles. <IMAGE>

b. At approximately what time is the car traveling the fastest? The slowest?

{Use of Tech} Flow from a tank A cylindrical tank ​​is full at time t=0 when a valve in the bottom of the tank is opened. By Torricelli’s law, the volume of water in the tank after t hours is V=100(200−t)², measured in cubic meters.

d. At what time is the magnitude of the flow rate a minimum? A maximum?  

{Use of Tech} Spring runoff The f​​low of a small stream is monitored for 90 days between May 1 and August 1. The total water that flows past a gauging station is given by v(t) = <matrix 2x2> where V is measured in cubic feet and t is measured in days, with t=0 corresponding to May 1.

c. Describe the flow of the stream over the 3-month period. Specifically, when is the flow rate a maximum?

Derivatives using tables Let $h\left(x\right)=f\left(g\right(x\left)\right)$ and $p\left(x\right)=g\left(f\right(x\left)\right)$. Use the table to compute the following derivatives.

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a. $h^{\prime }\left(3\right)$

Derivatives using tables Let $h(x)=f(g(x))$ and $p(x)=g(f(x))$. Use the table to compute the following derivatives.

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b. $h^{\prime }\left(2\right)$

Derivatives using tables Let $h(x)=f(g(x))$ and $p(x)=g(f(x))$. Use the table to compute the following derivatives.

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c. $p^{\prime }\left(4\right)$

Derivatives using tables Let $h(x)=f(g(x))$ and $p(x)=g(f(x))$. Use the table to compute the following derivatives.

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d. $p^{\prime }\left(2\right)$

Derivatives using tables Let $h(x)=f(g(x))$ and $p(x)=g(f(x))$. Use the table to compute the following derivatives.

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e. $h^{\prime }\left(5\right)$

27–76. Calculate the derivative of the following functions.

$y={(3x^2+7x)}^{10}$

27–76. Calculate the derivative of the following functions.

$y={(x^2+2x+7)}^8$

27–76. Calculate the derivative of the following functions.

$y=\sqrt{10x+1}$

27–76. Calculate the derivative of the following functions.

$\sqrt[3]{x^2+9}$

Vibrations of a spring Suppose an object of mass m is attached to the end of a spring hanging from the ceiling. The mass is at its equilibrium position $y=0$ when the mass hangs at rest. Suppose you push the mass to a position $y_0$ units above its equilibrium position and release it. As the mass oscillates up and down (neglecting any friction in the system), the position y of the mass after t seconds is $y=y_0\cos \left(t\sqrt{\frac{k}{m}}\right)$, where $k>0$ is a constant measuring the stiffness of the spring (the larger the value of $k$, the stiffer the spring) and $y$ is positive in the upward direction.

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Use equation (4) ​to answer the following questions.

c. How would the velocity be affected if the experiment were repeated with a spring having four times the stiffness ($k$ is increased by a factor of $4$)?

{Use of Tech} A damped oscillator The displacement of a mass on a spring suspended from the ceiling is given by $y=10e^{-\frac{t}{2}}\cos \left(\frac{\pi t}{8}\right)$.

a. Graph the displacement function. 

109-112 {Use of Tech} Calculating limits​ The following limits are the derivatives of a composite function g at a point a.

b. Use the Chain Rule to find each limit. Verify your answer by using a calculator.

$\underset{x\to 2}{\lim }\frac{{(x^2-3)}^5-1}{x-2}$

109-112 {Use of Tech} Calculating limits​ The following limits are the derivatives of a composite function g at a point a.

b. Use the Chain Rule to find each limit. Verify your answer by using a calculator.

$\underset{x\to 0}{\lim }\frac{\sqrt{4+\sin \left(x\right)}-2}{x}$

109-112 {Use of Tech} Calculating limits​ The following limits are the derivatives of a composite function g at a point a.

b. Use the Chain Rule to find each limit. Verify your answer by using a calculator.

$\underset{h\to 0}{\lim }\frac{\frac{1}{3{({(1+h)}^5+7)}^{10}}-\frac{1}{3{\left(8\right)}^{10}}}{h}$

{Use of Tech} Hours of daylight The number o​​f hours of daylight at any point on Earth fluctuates throughout the year. In the Northern Hemisphere, the shortest day is on the winter solstice and the longest day is on the summer solstice. At 40° north latitude, the length of a day is approximated by D(t) = 12−3 cos (2π(t+10) / 365), where D is measured in hours and 0≤t≤365 is measured in days, with t=0 corresponding to January 1.

b. Find the rate at which the daylight function changes.

{Use of Tech} A mixing tank A 500-liter (L) tank is filled with pure water. At time t=0, a salt solution begins flowing into the tank at a rate of 5 L/min. At the same time, the (fully mixed) solution flows out of the tank at a rate of 5.5 L/min. The mass of salt in grams in the tank at any time t≥0 is given by M(t) = 250(1000−t)(1−10−³⁰(1000−t)¹⁰) and the volume of solution in the tank is given by V(t) = 500-0.5t.

b. Graph the volume function and verify that the tank is empty when t=1000 min. 

{Use of Tech} Power and ​energy ​The total energy in megawatt-hr (MWh) used by a town is given by E(t) = 400t+2400/π sin πt/12, where t≥0 is measured in hours, with t=0 corresponding to noon.

b. At what time of day is the rate of energy consumption a maximum? What is the power at that time of day?

The Chain Rule for second derivatives

b. Use the formula in part (a) to calculate $\frac{d^2}{d{x}^2}(\sin (3x^4+5x^2+2))$.

Deriving​​ trigonometric identities

a. Differentiate both sides of the identity cos 2t = cos² t−sin² t to prove that sin 2 t= 2 sin t cos t.

Deriving​​ trigonometric identities

b. Verify that you obtain the same identity for sin2t as in part (a) if you differentiate the identity cos 2t = 2 cos² t−1.

Deriving​​ trigonometric identities

c. Differentiate both sides of the identity sin 2t = 2 sin t cost to prove that cos 2t = cos²t−sin²t.