Calculus Review

How do I know where f is increasing? It is where f prime is positive. 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? It is where f double prime equals zero or is undefined and the sign changes. The f prime function changes from increasing to decreasing or vise-versa. It is where f prime has maximums or minimums.

How do I know if the particle is moving to the left? It is where f prime is negative. 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? Find v(t) and a(t): If they have the same sign the particle is speeding up. If they have different signs the particle is slowing down.

What is speed? It is the absolute value of velocity.

What do I do if the problem says find the particular solution y = f(x)? This is asking you to find the original function that represents f. You are probably doing a separable variable problem.

What does 𝑑𝑦 𝑑𝑥 mean? This is asking for the first derivative. It could also be written 𝑑 𝑑𝑥

What does 𝑑 2 𝑦 𝑑𝑥² mean? This is asking for the second derivative. It could also be written 𝑑 2 𝑑 𝑥 2 .

What is the limit definition of a derivative? Or 𝑓 ′ 𝑎 = lim 𝑥→𝑎 𝑓 𝑥 −𝑓(𝑎) 𝑥−𝑎

What are the limit rules? Is it a hidden derivative? If it’s approaching infinity and it’s a polynomial over a polynomial then use the horizontal asymptote rules. Can you factor to simplify and just plug in the numbers. L’Hopital’s rule: do f(x) and g(x) both approach 0 or ±∞, then lim 𝑥→𝑎 𝑓(𝑥) 𝑔(𝑥) = lim 𝑥→𝑎 𝑓′(𝑥) 𝑔′(𝑥) .

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 2 1+4𝑥²

Differentiate: 1 2𝑒 𝑥 - 1 2 𝑒 −𝑥

Differentiate: 2 3𝑥 3× 2 3𝑥 ×ln⁡(2)

𝑦= 1 𝑥 2 +1 , Find 𝑑𝑦 𝑑𝑥 −2𝑥 𝑥 2 +1 −2

𝑓 𝑥 = 𝑥−1 𝑥 2 +1 4 , find 𝑓 ′ 𝑥 . 𝑥 2 +1 3 9 𝑥 2 −8𝑥+1

sin 2𝑥 + cos 3𝑥 𝑑𝑥= −cos⁡(2𝑥) 2 + sin⁡(3𝑥) 3 + C

𝑥 𝑥 2 −6 dx= 1 2 ln⃒ 𝑥 2 −6⃒+𝐶

𝑓 ′′ 𝑥 = 𝑥 2 𝑥−4 𝑥−8 . Find the x-coordinate(s) for points of inflection on f. x=4 and x=8

cos(xy)=x, find 𝑑𝑦 𝑑𝑥 − csc 𝑥𝑦 −𝑦 𝑥

Find the slope of the line tangent to the curve y=arctan(3x) at x = 1 3 . 3 2

Day Two Practice

Use a right Riemann sum with the four subintervals indicated by the data in the table to approximate 1 20 0 20 𝑊 𝑡 𝑑𝑡. t (minutes) 5 9 12 20 W(t) degrees F 54.0 58.2 63.1 68.1 70 1 20 5 58.2 +4 63.1 +3 68.1 +8(70) = 65.385

Is the previous estimate an overestimate or an underestimate? It is an overestimate, because the function is always increasing and the right Riemann sum would be above the curve.

5 𝑥 2 −2𝑥𝑦=𝑐𝑜𝑠𝑥, find 𝑑𝑦 𝑑𝑥 . 𝑑𝑦 𝑑𝑥 = − sin 𝑥 +2𝑦−10𝑥 −2𝑥

𝑑𝑦 𝑑𝑥 = −𝑥 𝑦 2 2 Find the solution y = f(x) to the given differential equation with the initial condition f(-1) = 2. 𝑦= 4 𝑥 2 +1

𝑑𝑦 𝑑𝑥 = −𝑥 𝑦 2 2 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⁡(3𝑥) 3 +𝐶

lim 𝑥→0 2𝑥 4 +5 𝑥 3 5𝑥 4 + 3𝑥 3 5 3

lim 𝑥→0 𝑠𝑖𝑛𝑥 𝑥 = 1

𝑓 𝑥 = 𝑥 2 −4𝑥+3𝑓𝑜𝑟𝑥≤2 𝑘𝑥+1𝑓𝑜𝑟𝑥>2 The function f is defined above 𝑓 𝑥 = 𝑥 2 −4𝑥+3𝑓𝑜𝑟𝑥≤2 𝑘𝑥+1𝑓𝑜𝑟𝑥>2 The function f is defined above. For what value of k, if any is f continuous at x = 2? k = -1

X=0 Where are the minimums? X=-1.5 and x = 6 The function f given by 𝑓 𝑥 = 1 2 𝑥 4 −3 𝑥 3 −9 𝑥 2 has a relative maximum at x = ? X=0 Where are the minimums? X=-1.5 and x = 6

What is the slope of the line tangent to the graph of 𝑦= 𝑒 −𝑥 𝑥+3 at x=1? − 5 16𝑒

lim ℎ→0 𝑒 3+ℎ − 𝑒 3 ℎ = 𝑒 3

lim 𝑥→∞ 4𝑥 6 −5𝑥 3𝑥 6 + 7𝑥 2 = 4 3

𝑑𝑦 𝑑𝑥 0 𝑥 2 sin 2𝑡 𝑑𝑡= 2𝑥(𝑠𝑖𝑛 2 𝑥 2 )