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 to find where f prime is positive or negative.

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.

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.

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.

What is speed?

OIt is the absolute value of velocity.

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.

What does mean?

OThis is asking for the first derivative.

OIt could also be written

What does mean?O This is asking for the second

derivative.O It could also be written .

What is the limit definition of a derivative?

OOr

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.

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).

Day One Practice

Differentiate: arctan 2x

21+4 𝑥 ²

Differentiate:

-

3×23𝑥×ln (2)

Differentiate:

,Find

−2 𝑥 (𝑥2+1 )− 2

, find

(𝑥2+1 )3 (9𝑥2−8 𝑥+1 )

∫ (sin (2𝑥 )+cos (3 𝑥 ) )𝑑𝑥=¿ ¿ + C

∫ 𝑥𝑥2−6

dx=¿

ln ⃒

. Find the x-coordinate(s) for points

of inflection on f.

x=4 and x=8

cos(xy)=x, find

−csc (𝑥𝑦 )− 𝑦𝑥

Find the slope of the line tangent to the curve y=arctan(3x) at x = .

32

Day Two Practice

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

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.

, find .

𝑑𝑦𝑑𝑥

=− sin (𝑥 )+2 𝑦−10𝑥

−2𝑥

Find the solution y = f(x) to the given differential

equation with the initial condition f(-1) = 2.

𝑦=4

𝑥2+1

Write an equation for the line tangent to the graph of f

at x = -1 if f(-1)=2.

y – 2 = 2(x + 1)

Day Three Practice

∫ sin (3𝑥 )𝑑𝑥=¿−cos (3 𝑥)

3+𝐶

lim𝑥→ 0

2𝑥4+5𝑥3

5𝑥4+3 𝑥3

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The function f is defined above. For what value of k, if any is f continuous at

x = 2?

k = -1

The function f given by has a relative

maximum at x = ?

X=0Where are the minimums?

X=-1.5 and x = 6

What is the slope of the line tangent to the

graph of

−5

16𝑒𝑥

=

𝑒3

=

43

𝑑𝑦𝑑𝑥∫0

𝑥2

sin (√2 𝑡 )𝑑𝑡=¿¿

2 𝑥(𝑠𝑖𝑛√2 𝑥2)