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$f\left(x\right)=8x-3;{x}_{1}=\mathrm{-1},{x}_{2}=3$
$f\left(x\right)=\text{\u2212}{x}^{2}+x+2;{x}_{1}=0.5,{x}_{2}=1.5$
$f\left(x\right)=\frac{4}{3x-1};{x}_{1}=1,{x}_{2}=3$
$-\frac{3}{4}$
$f\left(x\right)=\frac{x-7}{2x+1};{x}_{1}=\mathrm{-2},{x}_{2}=0$
$f\left(x\right)=\sqrt{x-9};{x}_{1}=10,{x}_{2}=13$
$f\left(x\right)={x}^{1\text{/}3}+1;{x}_{1}=0,{x}_{2}=8$
$0.25$
$f\left(x\right)=6{x}^{2\text{/}3}+2{x}^{1\text{/}3};{x}_{1}=1,{x}_{2}=27$
For the following functions,
$f\left(x\right)=\frac{x}{5}+6,a=\mathrm{-1}$
$f\left(x\right)=1-x-{x}^{2},a=0$
$f\left(x\right)=\frac{7}{x},a=3$
a. $\frac{\mathrm{-7}}{9}$ b. $y=\frac{\mathrm{-7}}{9}x+\frac{14}{3}$
$f\left(x\right)=\sqrt{x+8},a=1$
$f\left(x\right)=2-3{x}^{2},a=\mathrm{-2}$
a. $12$ b. $y=12x+14$
$f\left(x\right)=\frac{\mathrm{-3}}{x-1},a=4$
$f\left(x\right)=\frac{2}{x+3},a=\mathrm{-4}$
a. $\mathrm{-2}$ b. $y=\mathrm{-2}x-10$
$f\left(x\right)=\frac{3}{{x}^{2}},a=3$
For the following functions $y=f\left(x\right),$ find ${f}^{\prime}\left(a\right)$ using [link] .
$f\left(x\right)=\mathrm{-7}x+1,a=3$
$f\left(x\right)=3{x}^{2}-x+2,a=1$
$f\left(x\right)=\sqrt{x-2},a=6$
$f\left(x\right)=\frac{1}{x-3},a=\mathrm{-1}$
$f\left(x\right)=\frac{1}{\sqrt{x}},a=4$
For the following exercises, given the function $y=f\left(x\right),$
[T] $f\left(x\right)={x}^{2}+3x+4,P\left(1,8\right)$ (Round to $6$ decimal places.)
x | Slope ${m}_{PQ}$ | x | Slope ${m}_{PQ}$ |
---|---|---|---|
1.1 | (i) | 0.9 | (vii) |
1.01 | (ii) | 0.99 | (viii) |
1.001 | (iii) | 0.999 | (ix) |
1.0001 | (iv) | 0.9999 | (x) |
1.00001 | (v) | 0.99999 | (xi) |
1.000001 | (vi) | 0.999999 | (xii) |
a. $\text{(i)}\phantom{\rule{0.2em}{0ex}}5.100000,$ $\text{(ii)}\phantom{\rule{0.2em}{0ex}}5.010000,$ $\text{(iii)}\phantom{\rule{0.2em}{0ex}}5.001000,$ $\text{(iv)}\phantom{\rule{0.2em}{0ex}}5.000100,$ $\text{(v)}\phantom{\rule{0.2em}{0ex}}5.000010,$ $\text{(vi)}\phantom{\rule{0.2em}{0ex}}5.000001,$ $\text{(vii)}\phantom{\rule{0.2em}{0ex}}4.900000,$ $\text{(viii)}\phantom{\rule{0.2em}{0ex}}4.990000,$ $\text{(ix)}\phantom{\rule{0.2em}{0ex}}4.999000,$ $\text{(x)}\phantom{\rule{0.2em}{0ex}}4.999900,$ $\text{(xi)}\phantom{\rule{0.2em}{0ex}}4.999990,$ $\text{(x)}\phantom{\rule{0.2em}{0ex}}4.999999$ b. ${m}_{\text{tan}}=5$ c. $y=5x+3$
[T] $f\left(x\right)=\frac{x+1}{{x}^{2}-1},P\left(0,\mathrm{-1}\right)$
x | Slope ${m}_{PQ}$ | x | Slope ${m}_{PQ}$ |
---|---|---|---|
0.1 | (i) | $\mathrm{-0.1}$ | (vii) |
0.01 | (ii) | $\mathrm{-0.01}$ | (viii) |
0.001 | (iii) | $\mathrm{-0.001}$ | (ix) |
0.0001 | (iv) | $\mathrm{-0.0001}$ | (x) |
0.00001 | (v) | $\mathrm{-0.00001}$ | (xi) |
0.000001 | (vi) | $\mathrm{-0.000001}$ | (xii) |
[T] $f\left(x\right)=10{e}^{0.5x},P\left(0,10\right)$ (Round to $4$ decimal places.)
x | Slope ${m}_{PQ}$ |
---|---|
$\mathrm{-0.1}$ | (i) |
$\mathrm{-0.01}$ | (ii) |
$\mathrm{-0.001}$ | (iii) |
$\mathrm{-0.0001}$ | (iv) |
$\mathrm{-0.00001}$ | (v) |
−0.000001 | (vi) |
a. $\text{(i)}\phantom{\rule{0.2em}{0ex}}4.8771,$ $\text{(ii)}\phantom{\rule{0.2em}{0ex}}4.9875\phantom{\rule{0.2em}{0ex}}\text{(iii)}\phantom{\rule{0.2em}{0ex}}4.9988,$ $\text{(iv)}\phantom{\rule{0.2em}{0ex}}4.9999,$ $\text{(v)}\phantom{\rule{0.2em}{0ex}}4.9999,$ $\text{(vi)}\phantom{\rule{0.2em}{0ex}}4.9999$ b. ${m}_{\text{tan}}=5$ c. $y=5x+10$
[T] $f\left(x\right)=\text{tan}\phantom{\rule{0.1em}{0ex}}\left(x\right),P\left(\pi ,0\right)$
x | Slope ${m}_{PQ}$ |
---|---|
3.1 | (i) |
3.14 | (ii) |
3.141 | (iii) |
3.1415 | (iv) |
3.14159 | (v) |
3.141592 | (vi) |
[T] For the following position functions $y=s\left(t\right),$ an object is moving along a straight line, where $t$ is in seconds and $s$ is in meters. Find
$s\left(t\right)=\frac{1}{3}t+5$
a. $\frac{1}{3};$ b. $\text{(i)}\phantom{\rule{0.2em}{0ex}}0.\stackrel{\u2013}{3}$ m/s, $\text{(ii)}\phantom{\rule{0.2em}{0ex}}0.\stackrel{\u2013}{3}$ m/s, $\text{(iii)}\phantom{\rule{0.2em}{0ex}}0.\stackrel{\u2013}{3}$ m/s, $\text{(iv)}\phantom{\rule{0.2em}{0ex}}0.\stackrel{\u2013}{3}$ m/s; c. $0.\stackrel{\u2013}{3}=\frac{1}{3}$ m/s
$s\left(t\right)={t}^{2}-2t$
$s\left(t\right)=2{t}^{3}+3$
a. $2\left({h}^{2}+6h+12\right);$ b. $\text{(i)}\phantom{\rule{0.2em}{0ex}}25.22$ m/s, $\text{(ii)}\phantom{\rule{0.2em}{0ex}}24.12$ m/s, $\text{(iii)}\phantom{\rule{0.2em}{0ex}}24.01$ m/s, $\text{(iv)}\phantom{\rule{0.2em}{0ex}}24$ m/s; c. $24$ m/s
$s\left(t\right)=\frac{16}{{t}^{2}}-\frac{4}{t}$
Use the following graph to evaluate a. ${f}^{\prime}\left(1\right)$ and b. ${f}^{\prime}\left(6\right).$
a. $1.25;$ b. $0.5$
Use the following graph to evaluate a. ${f}^{\prime}\left(\mathrm{-3}\right)$ and b. ${f}^{\prime}\left(1.5\right).$
For the following exercises, use the limit definition of derivative to show that the derivative does not exist at $x=a$ for each of the given functions.
$f\left(x\right)={x}^{1\text{/}3},x=0$
$\underset{x\to {0}^{-}}{\text{lim}}\frac{{x}^{1\text{/}3}-0}{x-0}=\underset{x\to {0}^{-}}{\text{lim}}\frac{1}{{x}^{2\text{/}3}}=\infty $
$f\left(x\right)={x}^{2\text{/}3},x=0$
$f\left(x\right)=\{\begin{array}{c}1,x<1\\ x,x\ge 1\end{array},x=1$
$\underset{x\to {1}^{-}}{\text{lim}}\frac{1-1}{x-1}=0\ne 1=\underset{x\to {1}^{+}}{\text{lim}}\frac{x-1}{x-1}$
$f\left(x\right)=\frac{\left|x\right|}{x},x=0$
[T] The position in feet of a race car along a straight track after $t$ seconds is modeled by the function $s\left(t\right)=8{t}^{2}-\frac{1}{16}{t}^{3}.$
a. $\text{(i)}\phantom{\rule{0.2em}{0ex}}61.7244$ ft/s, $\text{(ii)}\phantom{\rule{0.2em}{0ex}}61.0725$ ft/s $\text{(iii)}\phantom{\rule{0.2em}{0ex}}61.0072$ ft/s $\text{(iv)}\phantom{\rule{0.2em}{0ex}}61.0007$ ft/s b. At $4$ seconds the race car is traveling at a rate/velocity of $61$ ft/s.
[T] The distance in feet that a ball rolls down an incline is modeled by the function $s\left(t\right)=14{t}^{2},$ where t is seconds after the ball begins rolling.
Two vehicles start out traveling side by side along a straight road. Their position functions, shown in the following graph, are given by $s=f(t)$ and $s=g\left(t\right),$ where $s$ is measured in feet and $t$ is measured in seconds.
a. The vehicle represented by $f\left(t\right),$ because it has traveled $2$ feet, whereas $g\left(t\right)$ has traveled $1$ foot. b. The velocity of $f\left(t\right)$ is constant at $1$ ft/s, while the velocity of $g\left(t\right)$ is approximately $2$ ft/s. c. The vehicle represented by $g\left(t\right),$ with a velocity of approximately $4$ ft/s. d. Both have traveled $4$ feet in $4$ seconds.
[T] The total cost $C(x),$ in hundreds of dollars, to produce $x$ jars of mayonnaise is given by $C\left(x\right)=0.000003{x}^{3}+4x+300.$
[T] For the function $f\left(x\right)={x}^{3}-2{x}^{2}-11x+12,$ do the following.
a.
b.
$a\approx -1.361,2.694$
[T] For the function $f\left(x\right)=\frac{x}{1+{x}^{2}},$ do the following.
Suppose that $N\left(x\right)$ computes the number of gallons of gas used by a vehicle traveling $x$ miles. Suppose the vehicle gets $30$ mpg.
a. $N\left(x\right)=\frac{x}{30}$ b. $\sim 3.3$ gallons. When the vehicle travels $100$ miles, it has used $3.3$ gallons of gas. c. $\frac{1}{30}.$ The rate of gas consumption in gallons per mile that the vehicle is achieving after having traveled $100$ miles.
[T] For the function $f\left(x\right)={x}^{4}-5{x}^{2}+4,$ do the following.
[T] For the function $f\left(x\right)=\frac{{x}^{2}}{{x}^{2}+1},$ do the following.
a.
b.
$\mathrm{-0.028},\mathrm{-0.16},0.16,0.028$
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