calculus review. how do i know where f is increasing? o it is where f prime is positive. o find the...
TRANSCRIPT
![Page 1: Calculus Review. How do I know where f is increasing? O It is where f prime is positive. O Find the derivative and set equal to zero. Use test points](https://reader037.vdocuments.us/reader037/viewer/2022110211/56649ed05503460f94bde1b4/html5/thumbnails/1.jpg)
Calculus Review
![Page 2: Calculus Review. How do I know where f is increasing? O It is where f prime is positive. O Find the derivative and set equal to zero. Use test points](https://reader037.vdocuments.us/reader037/viewer/2022110211/56649ed05503460f94bde1b4/html5/thumbnails/2.jpg)
How do I know where f is increasing?
O It is where f prime is positive.O Find the derivative and set equal to
zero. Use test points to find where f prime is positive or negative.
![Page 3: Calculus Review. How do I know where f is increasing? O It is where f prime is positive. O Find the derivative and set equal to zero. Use test points](https://reader037.vdocuments.us/reader037/viewer/2022110211/56649ed05503460f94bde1b4/html5/thumbnails/3.jpg)
How do I know where f has inflection points?
O It is where f double prime equals zero or is undefined and the sign changes.
O The f prime function changes from increasing to decreasing or vise-versa.
O It is where f prime has maximums or minimums.
![Page 4: Calculus Review. How do I know where f is increasing? O It is where f prime is positive. O Find the derivative and set equal to zero. Use test points](https://reader037.vdocuments.us/reader037/viewer/2022110211/56649ed05503460f94bde1b4/html5/thumbnails/4.jpg)
How do I know if the particle is moving to the left?
O It is where f prime is negative.O Find where f prime equals zero.
Then check test points on f prime.
![Page 5: Calculus Review. How do I know where f is increasing? O It is where f prime is positive. O Find the derivative and set equal to zero. Use test points](https://reader037.vdocuments.us/reader037/viewer/2022110211/56649ed05503460f94bde1b4/html5/thumbnails/5.jpg)
How do I know if the particle is speeding up or
slowing down?
O Find v(t) and a(t): If they have the same sign the particle is speeding up.
O If they have different signs the particle is slowing down.
![Page 6: Calculus Review. How do I know where f is increasing? O It is where f prime is positive. O Find the derivative and set equal to zero. Use test points](https://reader037.vdocuments.us/reader037/viewer/2022110211/56649ed05503460f94bde1b4/html5/thumbnails/6.jpg)
What is speed?
OIt is the absolute value of velocity.
![Page 7: Calculus Review. How do I know where f is increasing? O It is where f prime is positive. O Find the derivative and set equal to zero. Use test points](https://reader037.vdocuments.us/reader037/viewer/2022110211/56649ed05503460f94bde1b4/html5/thumbnails/7.jpg)
What do I do if the problem says find the particular
solution y = f(x)?
O This is asking you to find the original function that represents f. You are probably doing a separable variable problem.
![Page 8: Calculus Review. How do I know where f is increasing? O It is where f prime is positive. O Find the derivative and set equal to zero. Use test points](https://reader037.vdocuments.us/reader037/viewer/2022110211/56649ed05503460f94bde1b4/html5/thumbnails/8.jpg)
What does mean?
OThis is asking for the first derivative.
OIt could also be written
![Page 9: Calculus Review. How do I know where f is increasing? O It is where f prime is positive. O Find the derivative and set equal to zero. Use test points](https://reader037.vdocuments.us/reader037/viewer/2022110211/56649ed05503460f94bde1b4/html5/thumbnails/9.jpg)
What does mean?O This is asking for the second
derivative.O It could also be written .
![Page 10: Calculus Review. How do I know where f is increasing? O It is where f prime is positive. O Find the derivative and set equal to zero. Use test points](https://reader037.vdocuments.us/reader037/viewer/2022110211/56649ed05503460f94bde1b4/html5/thumbnails/10.jpg)
What is the limit definition of a derivative?
OOr
![Page 11: Calculus Review. How do I know where f is increasing? O It is where f prime is positive. O Find the derivative and set equal to zero. Use test points](https://reader037.vdocuments.us/reader037/viewer/2022110211/56649ed05503460f94bde1b4/html5/thumbnails/11.jpg)
What are the limit rules?
O Is it a hidden derivative?O If it’s approaching infinity and it’s a
polynomial over a polynomial then use the horizontal asymptote rules.
O Can you factor to simplify and just plug in the numbers.
![Page 12: Calculus Review. How do I know where f is increasing? O It is where f prime is positive. O Find the derivative and set equal to zero. Use test points](https://reader037.vdocuments.us/reader037/viewer/2022110211/56649ed05503460f94bde1b4/html5/thumbnails/12.jpg)
How do I prove a function is continuous?
Show the left hand limit and the right hand limit are equal and are equal to f(x).
![Page 13: Calculus Review. How do I know where f is increasing? O It is where f prime is positive. O Find the derivative and set equal to zero. Use test points](https://reader037.vdocuments.us/reader037/viewer/2022110211/56649ed05503460f94bde1b4/html5/thumbnails/13.jpg)
Day One Practice
![Page 14: Calculus Review. How do I know where f is increasing? O It is where f prime is positive. O Find the derivative and set equal to zero. Use test points](https://reader037.vdocuments.us/reader037/viewer/2022110211/56649ed05503460f94bde1b4/html5/thumbnails/14.jpg)
Differentiate: arctan 2x
21+4 𝑥 ²
![Page 15: Calculus Review. How do I know where f is increasing? O It is where f prime is positive. O Find the derivative and set equal to zero. Use test points](https://reader037.vdocuments.us/reader037/viewer/2022110211/56649ed05503460f94bde1b4/html5/thumbnails/15.jpg)
Differentiate:
-
![Page 16: Calculus Review. How do I know where f is increasing? O It is where f prime is positive. O Find the derivative and set equal to zero. Use test points](https://reader037.vdocuments.us/reader037/viewer/2022110211/56649ed05503460f94bde1b4/html5/thumbnails/16.jpg)
3×23𝑥×ln (2)
Differentiate:
![Page 17: Calculus Review. How do I know where f is increasing? O It is where f prime is positive. O Find the derivative and set equal to zero. Use test points](https://reader037.vdocuments.us/reader037/viewer/2022110211/56649ed05503460f94bde1b4/html5/thumbnails/17.jpg)
,Find
−2 𝑥 (𝑥2+1 )− 2
![Page 18: Calculus Review. How do I know where f is increasing? O It is where f prime is positive. O Find the derivative and set equal to zero. Use test points](https://reader037.vdocuments.us/reader037/viewer/2022110211/56649ed05503460f94bde1b4/html5/thumbnails/18.jpg)
, find
(𝑥2+1 )3 (9𝑥2−8 𝑥+1 )
![Page 19: Calculus Review. How do I know where f is increasing? O It is where f prime is positive. O Find the derivative and set equal to zero. Use test points](https://reader037.vdocuments.us/reader037/viewer/2022110211/56649ed05503460f94bde1b4/html5/thumbnails/19.jpg)
∫ (sin (2𝑥 )+cos (3 𝑥 ) )𝑑𝑥=¿ ¿ + C
![Page 20: Calculus Review. How do I know where f is increasing? O It is where f prime is positive. O Find the derivative and set equal to zero. Use test points](https://reader037.vdocuments.us/reader037/viewer/2022110211/56649ed05503460f94bde1b4/html5/thumbnails/20.jpg)
∫ 𝑥𝑥2−6
dx=¿
ln ⃒
![Page 21: Calculus Review. How do I know where f is increasing? O It is where f prime is positive. O Find the derivative and set equal to zero. Use test points](https://reader037.vdocuments.us/reader037/viewer/2022110211/56649ed05503460f94bde1b4/html5/thumbnails/21.jpg)
. Find the x-coordinate(s) for points
of inflection on f.
x=4 and x=8
![Page 22: Calculus Review. How do I know where f is increasing? O It is where f prime is positive. O Find the derivative and set equal to zero. Use test points](https://reader037.vdocuments.us/reader037/viewer/2022110211/56649ed05503460f94bde1b4/html5/thumbnails/22.jpg)
cos(xy)=x, find
−csc (𝑥𝑦 )− 𝑦𝑥
![Page 23: Calculus Review. How do I know where f is increasing? O It is where f prime is positive. O Find the derivative and set equal to zero. Use test points](https://reader037.vdocuments.us/reader037/viewer/2022110211/56649ed05503460f94bde1b4/html5/thumbnails/23.jpg)
Find the slope of the line tangent to the curve y=arctan(3x) at x = .
32
![Page 24: Calculus Review. How do I know where f is increasing? O It is where f prime is positive. O Find the derivative and set equal to zero. Use test points](https://reader037.vdocuments.us/reader037/viewer/2022110211/56649ed05503460f94bde1b4/html5/thumbnails/24.jpg)
Day Two Practice
![Page 25: Calculus Review. How do I know where f is increasing? O It is where f prime is positive. O Find the derivative and set equal to zero. Use test points](https://reader037.vdocuments.us/reader037/viewer/2022110211/56649ed05503460f94bde1b4/html5/thumbnails/25.jpg)
Use a right Riemann sum with the four subintervals indicated by the data in the table to approximate
t (minutes) 0 5 9 12 20
W(t) degrees F
54.0 58.2 63.1 68.1 70
= 65.385
![Page 26: Calculus Review. How do I know where f is increasing? O It is where f prime is positive. O Find the derivative and set equal to zero. Use test points](https://reader037.vdocuments.us/reader037/viewer/2022110211/56649ed05503460f94bde1b4/html5/thumbnails/26.jpg)
Is the previous estimate an overestimate or an
underestimate?
OIt is an overestimate, because the function is always increasing and the right Riemann sum would be above the curve.
![Page 27: Calculus Review. How do I know where f is increasing? O It is where f prime is positive. O Find the derivative and set equal to zero. Use test points](https://reader037.vdocuments.us/reader037/viewer/2022110211/56649ed05503460f94bde1b4/html5/thumbnails/27.jpg)
, find .
𝑑𝑦𝑑𝑥
=− sin (𝑥 )+2 𝑦−10𝑥
−2𝑥
![Page 28: Calculus Review. How do I know where f is increasing? O It is where f prime is positive. O Find the derivative and set equal to zero. Use test points](https://reader037.vdocuments.us/reader037/viewer/2022110211/56649ed05503460f94bde1b4/html5/thumbnails/28.jpg)
Find the solution y = f(x) to the given differential
equation with the initial condition f(-1) = 2.
𝑦=4
𝑥2+1
![Page 29: Calculus Review. How do I know where f is increasing? O It is where f prime is positive. O Find the derivative and set equal to zero. Use test points](https://reader037.vdocuments.us/reader037/viewer/2022110211/56649ed05503460f94bde1b4/html5/thumbnails/29.jpg)
Write an equation for the line tangent to the graph of f
at x = -1 if f(-1)=2.
y – 2 = 2(x + 1)
![Page 30: Calculus Review. How do I know where f is increasing? O It is where f prime is positive. O Find the derivative and set equal to zero. Use test points](https://reader037.vdocuments.us/reader037/viewer/2022110211/56649ed05503460f94bde1b4/html5/thumbnails/30.jpg)
Day Three Practice
![Page 31: Calculus Review. How do I know where f is increasing? O It is where f prime is positive. O Find the derivative and set equal to zero. Use test points](https://reader037.vdocuments.us/reader037/viewer/2022110211/56649ed05503460f94bde1b4/html5/thumbnails/31.jpg)
∫ sin (3𝑥 )𝑑𝑥=¿−cos (3 𝑥)
3+𝐶
![Page 32: Calculus Review. How do I know where f is increasing? O It is where f prime is positive. O Find the derivative and set equal to zero. Use test points](https://reader037.vdocuments.us/reader037/viewer/2022110211/56649ed05503460f94bde1b4/html5/thumbnails/32.jpg)
lim𝑥→ 0
2𝑥4+5𝑥3
5𝑥4+3 𝑥3
53
![Page 33: Calculus Review. How do I know where f is increasing? O It is where f prime is positive. O Find the derivative and set equal to zero. Use test points](https://reader037.vdocuments.us/reader037/viewer/2022110211/56649ed05503460f94bde1b4/html5/thumbnails/33.jpg)
The function f is defined above. For what value of k, if any is f continuous at
x = 2?
k = -1
![Page 34: Calculus Review. How do I know where f is increasing? O It is where f prime is positive. O Find the derivative and set equal to zero. Use test points](https://reader037.vdocuments.us/reader037/viewer/2022110211/56649ed05503460f94bde1b4/html5/thumbnails/34.jpg)
The function f given by has a relative
maximum at x = ?
X=0Where are the minimums?
X=-1.5 and x = 6
![Page 35: Calculus Review. How do I know where f is increasing? O It is where f prime is positive. O Find the derivative and set equal to zero. Use test points](https://reader037.vdocuments.us/reader037/viewer/2022110211/56649ed05503460f94bde1b4/html5/thumbnails/35.jpg)
What is the slope of the line tangent to the
graph of
−5
16𝑒𝑥
![Page 36: Calculus Review. How do I know where f is increasing? O It is where f prime is positive. O Find the derivative and set equal to zero. Use test points](https://reader037.vdocuments.us/reader037/viewer/2022110211/56649ed05503460f94bde1b4/html5/thumbnails/36.jpg)
=
𝑒3
![Page 37: Calculus Review. How do I know where f is increasing? O It is where f prime is positive. O Find the derivative and set equal to zero. Use test points](https://reader037.vdocuments.us/reader037/viewer/2022110211/56649ed05503460f94bde1b4/html5/thumbnails/37.jpg)
=
43
![Page 38: Calculus Review. How do I know where f is increasing? O It is where f prime is positive. O Find the derivative and set equal to zero. Use test points](https://reader037.vdocuments.us/reader037/viewer/2022110211/56649ed05503460f94bde1b4/html5/thumbnails/38.jpg)
𝑑𝑦𝑑𝑥∫0
𝑥2
sin (√2 𝑡 )𝑑𝑡=¿¿
2 𝑥(𝑠𝑖𝑛√2 𝑥2)