Calculus BC Semester 1 Review- Comprehensive Guide

What This Guide Covers

This is your semester 1 Calculus BC review. If you're bombing tests or just need to organize what you actually learned, you're in the right place. We'll hit limits, derivatives, and basic integration—the stuff that makes up roughly 60% of the AP exam.

Prerequisites: You should already have seen this material. If you haven't, start with your textbook. This guide assumes you sat through the lectures and mostly retained nothing.

Limits and Continuity

Limits are the foundation. Everything in calculus builds on this single idea: what happens to a function as you get arbitrarily close to a point without actually hitting it.

How to Actually Find Limits

Three methods, in order of preference:

• Direct substitution — plug in the value. If it works, you're done. Most limits in your class will be this easy.
• Factoring — when direct substitution gives 0/0, factor and cancel. Then try again.
• Rationalizing — for limits with square roots, multiply by the conjugate.

The Three Types of Discontinuities

• Removable — the hole can be "filled in." The limit exists.
• Jump — the function jumps. Limit doesn't exist from either side.
• Infinite — vertical asymptote. Function goes to ±∞.

Intermediate Value Theorem: If f is continuous on [a,b] and k is between f(a) and f(b), then there's at least one c in [a,b] where f(c) = k. That's it. Don't overthink it.

Limits at Infinity

For rational functions, compare degrees:

• Degree of denominator > numerator → limit = 0
• Degrees equal → limit = ratio of leading coefficients
• Degree of numerator > denominator → limit = ±∞ or doesn't exist

Derivatives

Derivatives measure instantaneous rate of change. That's the whole concept. The notation varies—f'(x), dy/dx, d/dx[f(x)]—but they all mean the same thing.

The Definition (Memorize This)

f'(x) = lim(h→0) [f(x+h) - f(x)] / h

You'll rarely use this definition after this unit, but you need it for the AP exam FRQs. They will ask you to derive something using this formula.

Derivative Rules You Need Cold

• Power rule: d/dx[xⁿ] = nxⁿ⁻¹
• Product rule: (fg)' = fg' + f'g
• Quotient rule: (f/g)' = (g·f' - f·g') / g²
• Chain rule: d/dx[f(g(x))] = f'(g(x)) · g'(x)

Chain rule is where most people fail. If you have a composite function, you're using the chain rule. Outside derivative times inside derivative. Every time.

Trig Derivatives

You need these memorized:

• d/dx[sin x] = cos x
• d/dx[cos x] = -sin x
• d/dx[tan x] = sec²x
• d/dx[cot x] = -csc²x
• d/dx[sec x] = sec x tan x
• d/dx[csc x] = -csc x cot x

Implicit Differentiation

When you can't solve for y explicitly, differentiate everything with respect to x and treat y as a function of x. Every time you differentiate y, multiply by dy/dx.

Example: x² + y² = 25

2x + 2y(dy/dx) = 0

dy/dx = -x/y

Then solve for dy/dx. That's the whole process.

Derivatives of Inverse Functions

If y = f⁻¹(x), then (f⁻¹)'(a) = 1 / f'(f⁻¹(a))

In plain English: the derivative of an inverse at a point equals 1 over the derivative of the original function at the corresponding point.

Derivatives of Inverse Trig Functions

• d/dx[arcsin x] = 1/√(1-x²)
• d/dx[arccos x] = -1/√(1-x²)
• d/dx[arctan x] = 1/(1+x²)

Exponential and Logarithmic Derivatives

• d/dx[eˣ] = eˣ
• d/dx[ln x] = 1/x
• d/dx[aˣ] = aˣ · ln a
• d/dx[logₐ x] = 1 / (x · ln a)

Related Rates

Problems where two or more quantities are changing and they're related by an equation. Steps:

1. Draw a diagram
2. Define variables
3. Write the equation relating them
4. Differentiate implicitly with respect to time
5. Substitute known values and solve

Common setups: shadows, ladders, cones, spheres, circles. If it's a geometry problem with something shrinking or growing, it's probably related rates.

Applications of Derivatives

Extreme Values

Critical points occur where f'(x) = 0 or f'(x) is undefined. These are your candidates for max/min.

First Derivative Test:

• Left of critical point is negative, right is positive → local minimum
• Left is positive, right is negative → local maximum
• Same sign on both sides → not an extremum

Second Derivative Test:

• f''(c) > 0 → local minimum
• f''(c) < 0 → local maximum
• f''(c) = 0 → test fails, use first derivative test

Mean Value Theorem

If f is continuous on [a,b] and differentiable on (a,b), then there's at least one c where f'(c) = [f(b) - f(a)] / (b - a).

This one shows up constantly on the exam. The average rate of change over an interval equals the instantaneous rate at some point in that interval.

Optimization

Find the maximum or minimum of something in a word problem. Process:

1. Identify what you're optimizing
2. Write a function for it
3. Use given constraints to reduce to one variable
4. Find critical points
5. Verify it's actually a max or min

Box problems, fence problems, minimal surface area—these all follow this exact pattern.

Curve Sketching

To sketch a function, find:

• Intercepts
• Critical points and classify them
• Points where f'(x) = 0 (possible inflection points)
• Concavity using f''(x)
• Asymptotes

Concavity rules: f'' > 0 is concave up, f'' < 0 is concave down. Inflection points occur where concavity changes.

Motion Problems

Position s(t), velocity v(t) = s'(t), acceleration a(t) = v'(t) = s''(t).

• Object moving right/up when v(t) > 0
• Object moving left/down when v(t) < 0
• Object speeding up when v(t) and a(t) have the same sign
• Object slowing down when v(t) and a(t) have opposite signs

L'Hôpital's Rule

For limits that give 0/0 or ∞/∞, take the derivative of the numerator and denominator separately. Repeat until you get a determinate form.

This only works for those two indeterminate forms. Don't apply it everywhere.

Integration

Integration is the reverse of differentiation. Where derivatives break things apart, integrals put them back together.

Antiderivatives

You need to know these basic forms:

• ∫xⁿ dx = xⁿ⁺¹/(n+1) + C (when n ≠ -1)
• ∫eˣ dx = eˣ + C
• ∫1/x dx = ln|x| + C
• ∫cos x dx = sin x + C
• ∫sin x dx = -cos x + C
• ∫sec²x dx = tan x + C

The Definite Integral

The Fundamental Theorem of Calculus ties antiderivatives to area:

∫ₐᵇ f(x)dx = F(b) - F(a), where F is any antiderivative of f.

That's it. Find the antiderivative, evaluate at the bounds, subtract.

Integration Techniques

U-Substitution is the reverse of the chain rule. If you see a composite function, try letting u equal the inner part.

Steps:

1. Choose u (usually the inner function)
2. Find du
3. Substitute everything
4. Integrate
5. Substitute back

Integration by Parts for products of functions. Formula:

∫u dv = uv - ∫v du

Choose u using LIATE (Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential). Pick the first category that appears in your integrand.

Integration of Trig Functions

• ∫sin²x dx → use half-angle identity
• ∫cos²x dx → use half-angle identity
• ∫sin mx cos nx dx → use product-to-sum formulas

For sin²x and cos²x, remember: sin²x = (1 - cos(2x))/2 and cos²x = (1 + cos(2x))/2.

Riemann Sums and Area

Definite integrals approximate area under curves using rectangles:

• Left endpoint: use left side of each rectangle
• Right endpoint: use right side
• Midpoint: use middle of each interval
• Trapezoidal: average left and right, more accurate

As the number of rectangles approaches infinity, the Riemann sum approaches the definite integral.

Average Value

The average value of f on [a,b] is (1/(b-a)) ∫ₐᵇ f(x)dx.

Multiply the integral by 1/(b-a). That's the whole formula.

Area Between Curves

∫ₐᵇ (top function - bottom function) dx

Find where the curves intersect. Determine which is on top in each interval. Subtract and integrate.

Volume of Solids of Revolution

Two methods:

Disk/Washer Method:

Volume = ∫π[f(x)]² dx (revolving around x-axis)

For y-axis or other axes, adjust the radius accordingly.

Shell Method:

Volume = ∫2π(radius)(height) dx or dy

Use washers when integrating perpendicular to the axis of rotation. Use shells when integrating parallel to it.

Quick Reference: Common Formulas

How to Use This Guide

Step 1: Cover the table above. Can you derive everything on it from memory? If not, that's your starting point.

Step 2: Work through problems from each section. Don't read more—do more. Calculus is a skill. You learn it by practicing, not by reading guides.

Step 3: Focus on your weak areas. If you bomb related rates but crush optimization, spend your time accordingly.

Step 4: Take a full practice exam under timed conditions. Grade it. Find your patterns.

What to Focus On

If you're running low on study time, prioritize:

• Chain rule applications (you'll use this constantly)
• Related rates and optimization (FRQ favorites)
• U-substitution and integration by parts
• Volume calculations (disks and shells)
• Mean Value Theorem and its applications

These topics make up the majority of the free response section. Master these and you're looking at a 4 minimum.