MATH 0220 — Analytic Geometry and Calculus 1

MATH 0220 Study Guide

Analytic Geometry and Calculus 1

Calculus connects change and accumulation. This guide covers limits, derivatives, applications, integrals, and the additional integration techniques in Pitt's published Fall 2026 schedule. That schedule includes integration by parts and trigonometric integration near the end, so do not assume your course stops at substitution. Scope: Pitt Fall 2026 MATH 0220 schedule.

These are original notes and practice examples. Follow your instructor's schedule and permitted methods. Formulas are editable text: (∫_^)() means integral, (lim_)() means limit, and C is an arbitrary constant of integration. Use radians for trigonometric calculus.

Topic roadmap

1. Functions and algebra. Review domains, factoring, fractions, exponents, logarithms, inverse functions, and trigonometry. Many apparent calculus errors begin with algebra.

2. Limits and continuity. Determine what a function approaches; do not confuse a limit with the function's value. Compare left and right behavior.

3. Derivatives. Interpret a derivative as local rate of change and tangent slope, then use differentiation rules efficiently.

4. Applications. Translate related rates, optimization, and curve analysis into equations. The modeling step matters as much as differentiation.

5. Integration. Interpret definite integrals as signed accumulation and antiderivatives as families of functions.

6. Fundamental theorem. Connect differentiation to integration, including variable limits of integration.

7. Integration techniques. Recognize substitution, integration by parts, and selected trigonometric patterns.

Important vocabulary

Domain: Inputs for which a function is defined. / Range: Outputs actually attained by a function. / Limit: Value approached as an input approaches a point or infinity. / One-sided limit: Limit approaching from only smaller or only larger inputs. / Continuity at a: (lim_x→a)(ƒ(x))=ƒ(a), with both one-sided limits and the function value defined. / Derivative: Limit of a difference quotient; measures instantaneous change. / Differentiability: Existence of a finite derivative at the point under consideration. / Tangent line: Local linear approximation with slope given by the derivative. / Chain rule: Rule for differentiating a composition of functions. / Implicit differentiation: Differentiating an equation involving x and y without first solving for y. / Critical number: Domain point where ƒ(x)=0 or the derivative fails to exist. / Local extremum: Maximum or minimum relative to nearby domain points. / Absolute extremum: Greatest or least value over the entire specified domain. / Concavity: Direction of bending, tracked by the behavior of ƒ^′ and often by ƒ^″. / Inflection point: Point where concavity changes; ƒ(x)=0 alone is insufficient. / Mean Value Theorem: Under its hypotheses, some instantaneous slope equals the interval's average slope. / Antiderivative: Function whose derivative is the given function. / Definite integral: Limit of suitable sums representing signed accumulation. / Riemann sum: Approximation formed from function values multiplied by subinterval widths. / Substitution: Change of integration variable that reverses a chain-rule structure. / Integration by parts: Integration rule derived from the product rule. / Linearization: Tangent-line approximation near a chosen input.

Formula reference

Limits and derivative definitions

The derivative definition is ƒ(x)=(lim_h→0)((ƒ*(x+h)−ƒ(x))/h). / Average rate of change on [a,b]: (ƒ(b)−ƒ(a))/(b−a). / Tangent line at x=a: y=ƒ(a)+ƒ(a)*(x−a). / Standard limits: (lim_x→0)(sin(x)/x)=1 and (lim_x→0)((e^x−1)/x)=1. / L'Hôpital's rule: For eligible 0/0 or ∞/∞ quotients, examine (lim_x→a)(ƒ(x)/g(x)) under the theorem's differentiability and limit conditions. Do not apply it to an arbitrary quotient.

Derivative rules

Power rule: d()/d(x)*x^n=n*x^(n−1) where defined; the derivative of a constant is 0. / Linearity: (a*ƒ+b*g)^′=a*ƒ^′+b*g^′ for constants a and b. / Product and quotient rules: (ƒ*g)^′=ƒ^′*g+ƒ*g^′ and (ƒ/g)^′=(ƒ^′*g−ƒ*g^′)/(g^2), where g≠0. / Chain rule: d()/d(x)*ƒ*(g(x))=ƒ^′*(g(x))*g(x). / Trigonometric derivatives: d()/d(x)*sin(x)=cos(x), d()/d(x)*cos(x)=−sin(x), and d()/d(x)*tan(x)=sec^2(x). / Exponential derivatives: d()/d(x)*e^x=e^x and d()/d(x)*a^x=a^x*ln(a) for a>0. / Logarithmic derivative: d()/d(x)*ln(x)=1/x for x≠0. / Inverse trigonometric derivatives: d()/d(x)*arcsin(x)=1/√(,1−x^2) for |x|<1, and d()/d(x)*arctan(x)=1/(1+x^2). / Hyperbolic derivatives: d()/d(x)*sinh(x)=cosh(x) and d()/d(x)*cosh(x)=sinh(x). / For F(x,y)=0, d(y)/d(x)=−(F_x)/(F_y) where the needed derivatives exist and (F_y)≠0. / Newton's iteration: (x_n+1)=(x_n)−ƒ((x_n))/ƒ((x_n)). A starting guess can fail or converge to an unintended root.

Applications and integration

Linearization: L(x)=ƒ(a)+ƒ(a)*(x−a); differential: d(y)=ƒ(x)*d(x). / Mean Value Theorem: ƒ(c)=(ƒ(b)−ƒ(a))/(b−a) for some c∈(a,b), if ƒ is continuous on [a,b] and differentiable on (a,b). / Power antiderivative: (∫_^)(x^n*d(x))=(x^(n+1))/(n+1)+C for n≠−1; logarithmic antiderivative: (∫_^)(1/x*d(x))=ln(x)+C on an interval excluding zero. / Basic antiderivatives: (∫_^)(e^x*d(x))=e^x+C, (∫_^)(cos(x)*d(x))=sin(x)+C, and (∫_^)(sin(x)*d(x))=−cos(x)+C. / Further antiderivatives: (∫_^)(sec^2(x)*d(x))=tan(x)+C and (∫_^)(1/(1+x^2)*d(x))=arctan(x)+C. / Fundamental Theorem of Calculus: (∫_a^b)(ƒ(x)*d(x))=F(b)−F(a) when F^′=ƒ and the theorem's conditions hold. / Variable upper limit: d()/d(x)*(∫_a^x)(ƒ(t)*d(t))=ƒ(x) for continuous ƒ. / Chain rule for variable limits: d()/d(x)*(∫_a^g(x))(ƒ(t)*d(t))=ƒ*(g(x))*g(x). / Substitution: (∫_^)(ƒ*(g(x))*g(x)*d(x))=(∫_^)(ƒ(u)*d(u)), with u=g(x). / Integration by parts: (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u)). / Trigonometric identities: sin^2(x)+cos^2(x)=1, 1+tan^2(x)=sec^2(x), and sin^2(x)=(1−cos(2*x))/2. / Common substitutions: x=a*sin(θ) for √(,a^2−x^2), and x=a*tan(θ) for √(,a^2+x^2), with appropriate domains. / Average value on [a,b]: (ƒ_avg)=1/(b−a)*(∫_a^b)(ƒ(x)*d(x)).

Worked examples

Example 1 A removable discontinuity

Find (lim_x→3)((x^2−9)/(x−3)). Factor the numerator: x^2−9=(x−3)*(x+3). For x≠3, the expression equals x+3, so the limit is 6. The original expression is still undefined at x=3; simplifying nearby behavior does not retroactively define its value there.

Example 2 Chain rule

For y=(3*x^2+1)^4, the outer function is u^4 and the inner function is u=3*x^2+1. Therefore, y^′=4*(3*x^2+1)^3*(6*x)=24*x*(3*x^2+1)^3. Checking the inner derivative prevents the most common missing-factor error.

Example 3 Optimization

A rectangle has perimeter 40*m. Let its sides be x and y, so y=20−x and A=x*(20−x), with 0<x<20. Since A^′=20−2*x, the critical point is x=10. Also, A^″=−2, confirming a local maximum; comparing boundary behavior shows the greatest area is 100*m^2. The problem's physical domain is part of the solution.

Example 4 Integration by parts

Evaluate (∫_^)(x*e^x*d(x)). Set u=x and d(v)=e^x*d(x), giving d(u)=d(x) and v=e^x. Then (∫_^)(x*e^x*d(x))=x*e^x−(∫_^)(e^x*d(x))=e^x*(x−1)+C. Differentiate the result to verify it returns x*e^x.

Practice questions and answers

Differentiate ln(x^2+1). Answer: (2*x)/(x^2+1). / Find the tangent to y=x^2 at x=2. Answer: y=4*x−4. / Evaluate (∫_0^2)(3*x^2*d(x)). Answer: 8. / Differentiate (∫_0^x^2)(cos(t)*d(t)). Answer: 2*x*cos(x^2). / Does ƒ(a)=0 guarantee an extremum? No; ƒ(x)=x^3 at x=0 is a counterexample. / Find (∫_^)(2*x*cos(x^2)*d(x)). Answer: sin(x^2)+C. / A circle's radius grows at 2*cm/s. Find the area growth rate when r=3*cm. Answer: d(A)/d(t)=2*π*rd(r)/d(t)=12*π*cm^2/s.

Mistakes to catch

Cancelling terms across addition rather than cancelling common factors. / Assuming continuity guarantees differentiability; |x| at x=0 disproves this. / Forgetting endpoints when finding absolute extrema on a closed interval. / Treating an integral below the x-axis as positive geometric area. / Changing variables in a definite integral but keeping incompatible limits. / Using L'Hôpital's rule before checking the indeterminate form.

Suggested web content

• OpenStax Calculus Volume 1: Use the contents menu for functions, limits, derivatives, applications, and integration. Match by topic because chapter numbering differs from Pitt's text.

• MIT Single Variable Calculus: Use lecture segments and worked recitations for a second explanation and additional practice.

• Pitt calculus curriculum: Find current departmental guidelines and schedules.

• Pitt student resources: Look for departmental practice materials and exam preparation resources.

Review routine

Alternate computation with explanation. For a derivative, describe its units and graph meaning; for an integral, explain what accumulates. Check antiderivatives by differentiation and approximate numerical answers to catch scale errors. Keep theorem hypotheses next to theorem names on your review sheet.