calculus review. how do i know where f is increasing? o it is where f prime is positive. o find the...

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Calculus Review

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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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

,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

, 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

∫ (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

∫ 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

. 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

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

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

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

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

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

, 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

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

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

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

∫ 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

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

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

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

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

=

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

=

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

𝑑𝑦𝑑𝑥∫0

𝑥2

sin (√2 𝑡 )𝑑𝑡=¿¿

2 𝑥(𝑠𝑖𝑛√2 𝑥2)