Chapter 2: Instantaneous Rate of Change: The DerivativeContent check — Calculus (Guichard), Chapter 2: Instantaneous Rate of Change: The Derivative

Draw the graph of the function ${\displaystyle y=f(x)=\sqrt{169-x^2}}$ between $x=0$ and $x=13$⁠. Find the slope $\mathrm{\Delta }y/\mathrm{\Delta }x$ of the chord between the points of the circle lying over (a) $x=12$ and $x=13$⁠, (b) $x=12$ and $x=12.1$⁠, (c) $x=12$ and $x=12.01$⁠, (d) $x=12$ and $x=12.001$⁠. Now use the geometry of tangent lines on a circle to find (e) the exact value of the derivative $f^{\prime }(12)$⁠. Your answers to (a)–(d) should be getting closer and closer to your answer to (e).

Use geometry to find the derivative $f^{\prime }(x)$ of the function ${\displaystyle f(x)=\sqrt{625-x^2}}$ in the text for each of the following $x$⁠: (a) 20, (b) 24, (c) $-7$⁠, (d) $-15$⁠. Draw a graph of the upper semicircle, and draw the tangent line at each of these four points.

Draw the graph of the function $y=f(x)=1/x$ between $x=1/2$ and $x=4$⁠. Find the slope of the chord between (a) $x=3$ and $x=3.1$⁠, (b) $x=3$ and $x=3.01$⁠, (c) $x=3$ and $x=3.001$⁠. Now use algebra to find a simple formula for the slope of the chord between $(3,f(3))$ and $(3+\mathrm{\Delta }x,f(3+\mathrm{\Delta }x))$⁠. Determine what happens when $\mathrm{\Delta }x$ approaches 0. In your graph of $y=1/x$⁠, draw the straight line through the point $(\mathrm{3,1}/3)$ whose slope is this limiting value of the difference quotient as $\mathrm{\Delta }x$ approaches 0.

Find an algebraic expression for the difference quotient ${\displaystyle (f(1+\mathrm{\Delta }x)-f(1))/\mathrm{\Delta }x}$ when ${\displaystyle f(x)=x^2-(1/x)}$⁠. Simplify the expression as much as possible. Then determine what happens as $\mathrm{\Delta }x$ approaches 0. That value is $f^{\prime }(1)$⁠.

Draw the graph of ${\displaystyle y=f(x)=x^3}$ between $x=0$ and $x=1.5$⁠. Find the slope of the chord between (a) $x=1$ and $x=1.1$⁠, (b) $x=1$ and $x=1.001$⁠, (c) $x=1$ and $x=1.00001$⁠. Then use algebra to find a simple formula for the slope of the chord between $1$ and $1+\mathrm{\Delta }x$⁠. (Use the expansion ${\displaystyle {(A+B)}^3=A^3+3A^{2}B+3A{B}^2+B^3}$⁠.) Determine what happens as $\mathrm{\Delta }x$ approaches 0, and in your graph of ${\displaystyle y=x^3}$ draw the straight line through the point $(\mathrm{1,1})$ whose slope is equal to the value you just found.

Find an algebraic expression for the difference quotient $(f(x+\mathrm{\Delta }x)-f(x))/\mathrm{\Delta }x$ when $f(x)=mx+b$⁠. Simplify the expression as much as possible. Then determine what happens as $\mathrm{\Delta }x$ approaches 0. That value is $f^{\prime }(x)$⁠.

An object is traveling in a straight line so that its position (that is, distance from some fixed point) is given by this table:

Find the average speed of the object during the following time intervals: $[\mathrm{0,1}]$⁠, $[\mathrm{0,2}]$⁠, $[\mathrm{0,3}]$⁠, $[\mathrm{1,2}]$⁠, $[\mathrm{1,3}]$⁠, $[\mathrm{2,3}]$⁠. If you had to guess the speed at $t=2$ just on the basis of these, what would you guess?

Let ${\displaystyle y=f(t)=t^2}$⁠, where $t$ is the time in seconds and $y$ is the distance in meters that an object falls on a certain airless planet. Draw a graph of this function between $t=0$ and $t=3$⁠. Make a table of the average speed of the falling object between (a) 2 seconds and 3 seconds, (b) 2 seconds and 2.1 seconds, (c) 2 seconds and 2.01 seconds, (d) 2 seconds and 2.001 seconds. Then use algebra to find a simple formula for the average speed between time $2$ and time $2+\mathrm{\Delta }t$⁠. (If you substitute $\mathrm{\Delta }t=1,0.1,0.01,0.001$ in this formula you should again get the answers to parts (a)–(d).) Next, in your formula for average speed (which should be in simplified form) determine what happens as $\mathrm{\Delta }t$ approaches zero. This is the instantaneous speed. Finally, in your graph of ${\displaystyle y=t^2}$ draw the straight line through the point $(\mathrm{2,4})$ whose slope is the instantaneous velocity you just computed; it should of course be the tangent line.

If an object is dropped from an 80-meter high window, its height $y$ above the ground at time $t$ seconds is given by the formula ${\displaystyle y=f(t)=80-4.9t^2}$⁠. (Here we are neglecting air resistance; the graph of this function was shown in figure 1.0.1.) Find the average velocity of the falling object between (a) 1 seconds and 1.1 seconds, (b) 1 seconds and 1.01 seconds, (c) 1 seconds and 1.001 seconds. Now use algebra to find a simple formula for the average velocity of the falling object between 1 seconds and $1+\mathrm{\Delta }t$ seconds. Determine what happens to this average velocity as $\mathrm{\Delta }t$ approaches 0. That is the instantaneous velocity at time $t=1$ second (it will be negative, because the object is falling).

Compute the limits. If a limit does not exist, explain why.

${\displaystyle \underset{x\to 3}{\lim }\frac{x^2+x-12}{x-3}}$

Compute the limits. If a limit does not exist, explain why.

${\displaystyle \underset{x\to 1}{\lim }\frac{x^2+x-12}{x-3}}$

Compute the limits. If a limit does not exist, explain why.

${\displaystyle \underset{x\to -4}{\lim }\frac{x^2+x-12}{x-3}}$

Compute the limits. If a limit does not exist, explain why.

${\displaystyle \underset{x\to 2}{\lim }\frac{x^2+x-12}{x-2}}$

Compute the limits. If a limit does not exist, explain why.

${\displaystyle \underset{x\to 1}{\lim }\frac{\sqrt{x+8}-3}{x-1}}$

Compute the limits. If a limit does not exist, explain why.

${\displaystyle \underset{x\to 0^{+}}{\lim }\sqrt{\frac{1}{x}+2}-\sqrt{\frac{1}{x}}}$

Compute the limits. If a limit does not exist, explain why.

${\displaystyle \underset{x\to 2}{\lim }3}$

Compute the limits. If a limit does not exist, explain why.

${\displaystyle \underset{x\to 4}{\lim }3x^3-5x}$

Compute the limits. If a limit does not exist, explain why.

${\displaystyle \underset{x\to 0}{\lim }\frac{4x-5x^2}{x-1}}$

Compute the limits. If a limit does not exist, explain why.

${\displaystyle \underset{x\to 1}{\lim }\frac{x^2-1}{x-1}}$

Compute the limits. If a limit does not exist, explain why.

${\displaystyle \underset{x\to 0^{+}}{\lim }\frac{\sqrt{2-x^2}}{x}}$

Compute the limits. If a limit does not exist, explain why.

${\displaystyle \underset{x\to 0^{+}}{\lim }\frac{\sqrt{2-x^2}}{x+1}}$

Compute the limits. If a limit does not exist, explain why.

${\displaystyle \underset{x\to a}{\lim }\frac{x^3-a^3}{x-a}}$

Compute the limits. If a limit does not exist, explain why.

${\displaystyle \underset{x\to 2}{\lim }{(x^2+4)}^3}$

Compute the limits. If a limit does not exist, explain why.

${\displaystyle \underset{x\to 1}{\lim }\{\begin{array}{cc}x-5\hfill & x\ne 1\text{,}\hfill \\ 7\hfill & x=1\text{.}\hfill \end{array}}$

Compute the limits. If a limit does not exist, explain why.

${\displaystyle \underset{x\to 0}{\lim }x\sin \left(\frac{1}{x}\right)}$ (Hint: Use the fact that $|\sin a|\le 1$ for any real number $a$⁠. You should probably use the definition of a limit here.)

Compute the limits. If a limit does not exist, explain why.

Evaluate the expressions by reference to this graph:

Find the derivative of ${\displaystyle y=f(x)=\sqrt{169-x^2}}$⁠.

Find the derivative of ${\displaystyle y=f(t)=80-4.9t^2}$⁠.

Find the derivative of ${\displaystyle y=f(x)=x^2-(1/x)}$⁠.

Find the derivative of ${\displaystyle y=f(x)=a{x}^2+bx+c}$ (where $a$⁠, $b$⁠, and $c$ are constants).

Find the derivative of ${\displaystyle y=f(x)=x^3}$⁠.

Find the derivative of ${\displaystyle y=f(x)=2/\sqrt{2x+1}}$

Find the derivative of $y=g(t)=(2t-1)/(t+2)$

Find an equation for the tangent line to the graph of ${\displaystyle f(x)=5-x-3x^2}$ at the point $x=2$

Find a value for $a$ so that the graph of ${\displaystyle f(x)=x^2+ax-3}$ has a horizontal tangent line at $x=4$⁠.

Is $f(x)=\sin (x)$ a bounded function? If so, find the smallest $M$⁠.

Is ${\displaystyle s(t)=1/(1+t^2)}$ a bounded function? If so, find the smallest $M$⁠.

Is $v(u)=2\ln |u|$ a bounded function? If so, find the smallest $M$⁠.

Consider the function $$h(x)=\{\begin{array}{cc}2x-3,\hfill & \text{if }x<1\hfill \\ 0,\hfill & \text{if }x\ge 1\text{.}\hfill \end{array}$$ Show that it is continuous at the point $x=0$⁠. Is $h$ a continuous function?

Approximate a root of ${\displaystyle f=x^3-4x^2+2x+2}$ to one decimal place.

Approximate a root of ${\displaystyle f=x^4+x^3-5x+1}$ to one decimal place.