What is the $1000^{\text{th}}$ derivative of $\sin(x)$ ?

Derivatives of Sine and Cosine

3-68.

Summarize all of the tools you have developed so far to write the equation of a slope function.

3-69.

Use the definition of the derivative to write equations for $f^\prime (x)$ and $g^\prime (x)$ if $f(x)=\sin(x)$ and $g(x)=\cos(x)$ . You will need to use the trigonometric identities to simplify your expressions.

3-70.

DERIVATIVES OF SINE AND COSINE GRAPHICALLY

It appears that the derivatives of sine and cosine are related. What about the second derivatives?

Set up five sets of axes, making sure the y-axes of the graphs are vertically aligned. Each x-axis should have domain $−2\pi\le x\le2\pi$ , scaled by $\frac { \pi } { 6 }$ . Each y-axis should have range $−2\le y\le2$ , scaled by $\frac { 1 } { 2 }$ .
As accurately as you can, sketch $f(x)=\sin(x)$ on the first set of axes. Draw bold dots on all maximum and minimum points.
On the second set of axes, sketch $y = f^\prime (x)$ as accurately as you can. Compare the graph of $y = f(x)$ with $y = f^\prime (x)$ . What does f look like when $f ^\prime (x) = 0$ ?

Repeat the process for $f^{\prime \prime}$ , $f^{\prime \prime \prime}$ , and $f^{(4)}$ , the second, third, and fourth derivatives of $f$ . As you work, you might discover shortcuts that will expedite this process. What do you notice about the fourth derivative?

Notation: We cannot simply put more and more tick marks for, say, the 7^th derivative. Instead, the 7^th derivative of f is written $f^{(7)}(x)$ . The $n^{th}$ derivative of a function is written $f^{(n)}(x)$ .

Predict $f^{(20)}(x)$ and $f^{(101)}(x)$ .

3-71.

Rewrite $y = \frac { 1 } { x }$ using exponents.

Write the slope function $y^\prime$ , or $\frac { d y } { d x }$ , algebraically by using the definition of the derivative.
Use the Power Rule to confirm your answer to part (a).

3-72.

The graph of the equation $y = x^3 − 9x^2 − 16x + 1$ has a slope of $5$ at exactly two points. What are the coordinates of these points? Describe your process.

3-73.
Given each function below, write an equation for $f^\prime$ . Homework Help ✎
$f(x) = −x^3$
$f(x) = \frac { 1 } { x ^ { 2 } }$
$f(x) = \sqrt { 2 }$
$f(x)=3\sin(x+\pi)$
3-74.
Write and then compute a Riemann sum to estimate the area the curve for $0\le x\le8$ given $f(x) = \sqrt { 64 - x ^ { 2 } }$ . Choose the number of rectangles so that your answer will be a good approximation of the area. What is the name of the shape of which you calculated the area? Confirm the accuracy of the Riemann sum by calculating the area geometrically. 3-74 HW eTool. Homework Help ✎
3-75.
Differentiate the following expressions. Homework Help ✎
$\frac { d } { d x } ( 6 ^ { 3 } )$
$\frac { d } { d x } ( \frac { 2 } { 5 } x ^ { 15 } - \frac { 3 } { 4 } x ^ { 2 } )$
$\frac { d } { d t } ( t ^ { - 9 } )$
$\frac { d } { d m } ( m ^ { 3 / 4 } )$
3-76.
Compare three different methods to find a derivative of $f(x) = 2x^3 − x$ . Homework Help ✎
1. Use the definition of a derivative.
2. Use the Power Rule. Does your answer agree with your answer from part (a)?
3. Use your graphing calculator to graph $f^\prime (x)= \frac{f(x+h)-f(x)}{h}$ for $h = 0.01$ . Does the graph match that of your answer from part (a)?
3-77.
Lazy Lulu is looking at this limit: $\lim\limits _ { x \rightarrow 3 } \large\frac { x ^ { 3 } + x - 30 } { x - 3 }$ and does not want to calculate it using algebra. Lulu recognizes this limit as a definition of the derivative at a point. She thinks she can use the Power Rule instead. Homework Help ✎
1. What variation of the definition of the derivative is this?
2. What is the equation of $f$ ? What is the value of $a$ ?
3. Use the Power Rule to write an equation for $f^\prime$ and then use it to calculate $f^\prime(a)$ .
3-78.
The position of a ball as a function of time is given by the function below where $s(t)$ is in meters and $t$ is in seconds. Homework Help ✎
$s ( t ) = \sqrt { t + 1 }$
1. Use your calculator to approximate the instantaneous velocity of the ball at $1, 5, 10,$ and $100$ seconds.
2. What happens to the velocity of the ball after a very long time (i.e. as $t\rightarrow \infty$ )?
3. What happens to the position of the ball after a very long time, (i.e. what is)? Does this make sense given your answer to part (b)?
3-79.
Calculate the average velocity between $0$ and $50$ seconds for each of the graphs below. Homework Help ✎
3-80.
The graph at right shows the height of a soccer ball after being kicked straight up into the air. Homework Help ✎
Explain why the slope of the tangent at point A will determine the velocity of the ball at that point.
At which of the labeled points is the velocity the greatest? How can you tell?
At what point does the ball momentarily stop? What is the velocity at this point?
The distance from the ground can be described by the function $s(t) = −16t^2 + 76.8t + 5$ . Write an equation for the velocity if $v(t) = s^\prime(t)$ .
Calculate the instantaneous velocity at $t = 2, 3,$ and $4$ seconds.
What does negative velocity represent in this problem?
3-81.
Evaluate each limit. If the limit does not exist due to a vertical asymptote, then add an approach statement stating if y is approaching negative or positive infinity Homework Help ✎
$\lim\limits _ { x \rightarrow 7 ^ { + } } \large\frac { 3 x - 21 } { x ^ { 2 } - x - 42 }$
$\lim\limits _ { x \rightarrow - \infty } \large\frac { 2 - 3 x ^ { 2 } } { 2 x ^ { 2 } + 5 x - 7 }$
$\lim\limits _ { x \rightarrow 0 } \large\frac { \sqrt { x + 2 } - \sqrt { 2 } } { x }$
$\lim\limits _ { x \rightarrow \infty } \large\frac { x ^ { 2 } + 3 x } { 1 - x ^ { 2 } - x ^ { 3 } }$

Calculus Third Edition