Limits & Continuity
Derivatives & Tangent Lines
Applications of Derivatives
Integrals & Accumulation
Differential Equations & Motion
100

 What is the limit of lim⁡x→3(2x+1)

2(3)+1=7

100

Find the derivative:

f(x) = x^3 + 2x
f'(x) = 3x^2 + 2
100

What does the first derivative tell you about a function?

Increasing/decreasing behavior and slope of tangent line

100

Find the following integral:

∫ x^2 dx
∫ x^2 dx = (1/3)x^3 + C
100

If dy/dt=ky, what type of growth is this?

Exponential growth

200

Determine lim⁡x→0 sin⁡x/x

The limit is 1

200

Find the derivative:

f(x) = (2x^3)(x^2)
Use the product rule:  
f'(x) = (2x^3)'(x^2) + (2x^3)(x^2)'  
= 6x^2 * x^2 + 2x^3 * 2x  
= 6x^4 + 4x^4 = 10x^4
200

A function has a local minimum when...

The first derivative changes from negative to positive


200

Evaluate the definite integral:

∫₀³ 2x dx
∫₀³ 2x dx = [x²]₀³ = 9 - 0 = 9
200

Solve dy/dx=2x with initial condition y(0)=3

y=x2+C→3=0+C⇒C=3⇒y=x2+3

300

True or False: If a function is continuous, then it is always differentiable

False. A function can be continuous but not differentiable

300

Find the derivative:

f(x) = ln(x^2 + 1)
f'(x) = (1 / (x^2 + 1)) * 2x = 2x / (x^2 + 1)
300

Intervals where 𝑓(𝑥) = 𝑥3 − 3𝑥2increasing?

f′(x)=3x2−6x = 3x(x−2), Set 𝑓′( 𝑥 ) > 0→ increasing on (−∞,0)∪(2,∞)

300

Given f′(x) = 3x² and f(1) = 4, find f(x).

∫ 3x² dx = x³ + C  
f(1) = 4 → 1³ + C = 4 → C = 3  
So, f(x) = x³ + 3
300

If s(t)=3t2−12t+9 , when is the object at rest?

Find v(t)=s′(t)=6t−12
Set v(t)=0: t=2

400

Find the value of a that makes the function continuous at x = a:

f(x) = (x^2 - a^2)/(x - a)
f(x) = (x^2 - a^2)/(x - a) simplifies to x + a.

So, f(a) = a + a = 2a.

Final Answer: f(a) = 2a
400

Find the derivative using the quotient rule:

f(x) = (x^2 + 1) / (x - 3)
Use the quotient rule:  
f'(x) = [(x - 3)(2x) - (x^2 + 1)(1)] / (x - 3)^2  
= [(2x)(x - 3) - (x^2 + 1)] / (x - 3)^2  
= (2x^2 - 6x - x^2 - 1) / (x - 3)^2  
= (x^2 - 6x - 1) / (x - 3)^2
400

Find the critical points of the function:

f(x) = x^3 - 3x^2 + 4

and determine if each is a local maximum, local minimum, or neither.

Final Answer:  
Critical points at x = 0 (local max) and x = 2 (local min)
400

Estimate the area under the curve using the trapezoidal rule:

x = 0, 2, 4  
f(x) = 3, 5, 7
h = 2  
Area ≈ (1/2)(2)[3 + 2(5) + 7] = 20
400

Match the slope field to dy/dx=y

Exponential growth — all slopes increase as y increases; solutions curve upward.

500

Evaluate lim⁡x→0+ 1/x

+∞ — as x approaches 0 from the right, the function grows without bound

500

If f(x) = sin(x^2), find f'(x).

Use the chain rule:  
f'(x) = cos(x^2) * 2x = 2x cos(x^2)
500

A particle moves along a line so that its position at time t is given by:

s(t) = t^3 - 6t^2 + 9t

Find when the particle is at rest and determine whether it's speeding up or slowing down at t = 1.

at t = 1, the particle is at rest and slowing down
500

Find the net change in velocity if a(t) = cos(t) from t = 0 to t = π.

∫₀^π cos(t) dt = sin(t) |₀^π = sin(π) - sin(0) = 0
500

A population of bacteria grows at a rate proportional to its size.

Write a differential equation for the population P(t), and solve it given that P(0) = 100 and the population doubles in 5 hours.

 Final Answer: P(t) = 100e((ln(2)/5)t)

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