Learn Calculus 1 Through Example Problems

Master limits, derivatives, integrals, and applications through step-by-step example problems that build understanding, strengthen problem-solving skills, and prepare you for homework, quizzes, and exams.

Chapter 2

Chapter 3

Chapter 4

Chapter 5

Chapter 6

Chapter 2 - Limits and Derivatives

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2.1 & 2.2 - Tangent, Velocity, and the Limit of a Function

Find the equation of the tangent line to y= sqrt(x+4) at the point P(5,3).
Find average and instantaneous velocity of a rock launched upward with position 𝑦 = − 16t^2 + 96t.

2.3 - Calculating Limits Using Limit Laws

Evaluate limits of functions as x approaches a value using graphs and algebra.
Determine infinite limits and identify vertical asymptotes from graphs and expressions.
Apply limit laws to compute limits of sums, products, quotients, powers, and roots.
Solve limits with indeterminate forms like 0/0 and infinity minus infinity using algebra.

2.4 - The Precise Definition of a Limit

Evaluate limits with the Squeeze Theorem by bounding a function between two simpler functions.
Find delta for a given epsilon using the epsilon-delta definition of a limit.
Prove a limit statement using the epsilon-delta definition of a limit.

2.5 & 2.6 - Continuity and Limits at Infinity

State where each function is discontinuous and classify as removable, jump, or infinite discontinuity
Find the value of c that makes the piecewise function continuous at x=3.
Identify x values where f is discontinuous and explain why, then test continuity from the left or right
Determine the intervals where f(x) = sin(x)/(x+1), sqrt(x-4)(x^3+1), and ln(x-1)/(x^2-4) are continuous.
Use the Intermediate Value Theorem to show a root exists for x^3 - x - 2 and sqrt(x-5) = 1/(x+3)
Evaluate limits at infinity for (4x-3)/(2x-1), sqrt(2x^2+1)/(5x-3), x^2 - x, and sqrt(x^4+6x^2) - x^2.
Find horizontal and vertical asymptotes for y = (4x-3)/(2x-1) and y = (x^3 - x)/(x^2 - 6x + 5).

2.7 & 2.8 - Derivatives and Rates of Change

Find the equation of the tangent line to y = x - x^3 at the point (2, -6).
Find the instantaneous velocity when t=1 for s = -16t^2 + 96t, a rock launched upward at 96 ft/sec.
Find the derivative of f(x) = 3x^2 - 4x + 1 using the definition and evaluate limits that represent derivatives.
Find derivatives of f(x)=5x-4 and f(x)=sqrt(x) using the definition and state their domains.
Graph the derivative f’ of a function f from its graph for multiple examples.
From the graph of f, state all x-values where f is not differentiable.
Identify which curve represents f, f’, and f’’ from the three given graphs.

Chapter 3 - Differentiation Rules

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3.1 & 3.2 - Core Derivative Rules

Differentiate f(x)=x^65, f(t)=1/t, g(x)=sqrt(x), and y=x^(-1/3) using the power rule.
Find where y=2e^x-2x has a horizontal tangent, where y=e^x is parallel to y=2x, and show no tangent with slope 2 for y=2e^x+3x+5x^3.
Differentiate using the quotient rule: (x^2+1)/(x^3-1), sqrt(x)/(2+x), and h(x)=g(x)/x given g(2)=4, g’(2)=-4.
Differentiate using the product rule: x^3(2x^2+1), (3x-5)(4x^2-3), and h(x)=xg(x) given g(3)=5, g’(3)=2.
Differentiate using the product rule: x^3(2x^2+1), (3x-5)(4x^2-3), and h(x)=xg(x) given g(3)=5, g’(3)=2.

3.3 & 3.4- Trigonometric and Chain Rules

Evaluate limits as x→0: sin5x/3x, tan6x/sin2x, and (1−cosx)/sinx using fundamental trig limits.
Identify inner and outer functions for y=sqrt(1+x^5), y=(2x^3+5)^4, and y=sin(tanx).
Differentiate using the chain rule: (2x^4+5x^2)^3, (x^4-1)^3(x^3+1)^4, (x^2/(x+1))^5, and sin(cos(tanx)).
Differentiate x^2sinx, e^xcosx, secx tanx, and sinx/(1+cosx). Prove d/dx secx = secx tanx and find tangent line to y=sinx+cosx at (0,1)
Chain rule formula: derivative of outer times derivative of inner. Differentiate y=(2x^3+5)^4 using both forms.

3.5 - Implicit Differentiation and Inverse Trig Derivatices

Find dy/dx by implicit differentiation. Example: given x^2 + y^2 = 25, find dy/dx.
Differentiate implicitly: 1+x=sin(xy^2), sqrt(x+y)=1+x^2y^2, tan(x−y)=y/(1+x^2). Includes tangent line and second derivative problems.
Differentiate y=(tan^(-1)(4x))^2, y=sin^(-1)(3x+1), and y=x sec^(-1)(2x).

3.6 - Derivatives of Logarithmic Functions

Differentiate f(x)=ln(sinx), f(x)=x ln(x^2+1), f(x)=ln(3/x), and f(x)=log_5(2x^2+7).
Use logarithmic differentiation to find dy/dx for y=x(x^2+1)/sqrt(x+1) and y=x^sinx with step-by-step process.

3.7 & 3.8 - Rates of Change and Exponential Growth& Decay

Analyze motion for s(t)=t^3−9t^2+15t+10: velocity, rest, direction, distance, acceleration, and speed changes.
Find velocity, maximum height, and time to hit the ground for projectile h(t)=1.5+19.6t−4.9t^2.
Find the rate of change at t=10 minutes for concentration C(t)=0.05e^(-0.1t).
Find the rate of population growth at t=4 hours for P(t)=5000/(1+9e^(-0.6t)).
Find the marginal cost when 200 items are produced for C(x)=5000+10x+0.01x^2.
Exponential population growth of bacteria from 150 to 600 in 2 hours. Find growth model, population after 4 hours, rate, and time to 12,000.
Radioactive decay of iodine-131 with half-life 8 days. Find remaining mass, amount after 24 days, and time for 5 mg.

3.9 - Related Rates

Related rates problem: Oil spill spreads in a circular pattern. Radius increases at 1 m/s. Find the rate the area increases when the radius is 30 m.
Related rates problem: A spherical balloon’s radius increases at 4 mm/s. Find the rate the volume is increasing when the diameter is 80 mm.
Related rates problem: Water is pumped into an inverted conical tank at 2 m³/min. Tank height is 6 m, radius 4 m. Find how fast the water level rises when the depth is 2 m.
Related rates problem: A 10-ft ladder rests against a vertical wall, bottom slides away at 2 ft/s. Find how fast the top slides down when the bottom is 6 ft from the wall.
Related rates problem: A 15-ft streetlight casts a shadow of a 6-ft tall person walking away at 4 ft/s. Find the rate the tip of the shadow moves when the person is 30 ft from the pole.
Related rates problem: A rocket rises vertically at 40 m/s. An observer is 60 m away from the launch pad. Find the rate the angle of elevation increases when the rocket is 80 m high.
Related rates problem: Car A travels north at 45 mph, Car B east at 60 mph. Find the rate the distance between them increases 2 hours after leaving the same intersection.
Related rates problem: A boat is pulled toward a dock by a rope attached 5 m above water. Rope is reeled in at 2 m/s. Find the rate the boat approaches when 13 m of rope is out.

3.10 - Linear Approximations and Differentials

Examples finding linearization of functions including f(x) = x³ – 3x² + 4 at a = 1 and f(x) = sin x at a = π/6, with graphs of f(x) and L(x).
Problem estimating cos(31.5°) using linearization L(x) = f(a) + f′(a)(x – a) and differentials dy = f′(x) dx.
Example using differentials for y = √x at x = 1 and dx = 1, comparing dy = f′(x) dx with actual change Δy.
Graph showing accuracy of linear approximation for f(x) = √(x + 3) at a = 1, comparing linear estimate and actual curve.
Example estimating maximum, relative, and percentage error for the area of a circle with radius 24 cm and measurement error 0.2 cm using dy = f′(x) dx.

3.11 - Hyperbolic Functions

Graph showing cosh x and sinh x curves with example problems finding sinh 0, cosh 0, sinh 1, cosh 1, and tanh 1 using exponential definitions.
List of key hyperbolic identities including cosh²x – sinh²x = 1, 1 – tanh²x = sech²x, and addition formulas for sinh(x + y) and cosh(x + y).
Table of hyperbolic derivative formulas and examples such as f(x) = sinh(x²), f(x) = sinh²x, and f(x) = eˣ cosh x.
Examples finding numerical values and derivatives of inverse hyperbolic functions including sinh⁻¹(1), cosh⁻¹(1), tanh⁻¹(0), sinh⁻¹(tan x), cosh⁻¹(√x), and coth⁻¹(sec x).

Chapter 4 - Applications of Differentiation

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4.1 & 4.2 - Max and Min Values and the Mean Value Theorem

Graph illustrating local and absolute maximum and minimum points of a function with labeled coordinates showing where extrema occur.
Steps for finding critical numbers by setting f′(x) = 0 or checking where f′(x) is undefined, with examples including polynomial, radical, and trigonometric functions.
Examples showing how to find absolute maximum and minimum values of functions such as x³ – 6x² + 9x + 2 and x – 2cos x on given intervals.
Examples applying Rolle’s Theorem to find values of c where f′(c) = 0, including cubic and rational functions, with a graph showing f(a) = f(b).
Examples using the Mean Value Theorem to find points c satisfying the theorem and estimating function values, with a graph showing secant and tangent line slopes.

4.3 - How Derivatives Affect the Shape of a Graph

Graph and examples illustrating how to find intervals where a function is increasing or decreasing using the first derivative.
Graph showing concave up, concave down, and inflection points, with examples identifying intervals of concavity using the second derivative.
Diagrams and examples showing how to determine local maximum and minimum values by analyzing sign changes in f′(x).
Visual explanation and examples applying the second derivative test to determine local maxima and minima based on the sign of f″(x).

4.4 - Indeterminate Forms and L’Hospital’s Rule

Examples of limits with indeterminate forms such as 0/0 and ∞/∞, demonstrating how to apply L’Hôpital’s Rule to evaluate each case.
Explanation of indeterminate powers like 0⁰, ∞⁰, and 1∞, with an example showing how to evaluate (cos 3x)^(5/x) using logarithmic transformation and L’Hôpital’s Rule.

4.5 - Summary of Curve Sketching

Step-by-step process for sketching a polynomial function, showing how to analyze domain, intercepts, symmetry, end behavior, intervals of increase/decrease, extrema, and concavity.
Guided approach to sketching a rational function, including finding domain, intercepts, symmetry, horizontal asymptotes, and analyzing intervals and concavity.
Example of sketching a rational function that has a slant asymptote, illustrating how the graph approaches a diagonal line as x increases or decreases.
Graph and steps for sketching a rational function with vertical asymptotes, showing how asymptotic behavior affects the curve near undefined values.

4.7 - Optimization Problems

Example demonstrating how to find two numbers with a fixed difference whose product is a minimum using derivative-based optimization.
Example finding the rectangle with a fixed area that has the smallest possible perimeter using calculus optimization methods.
Example of maximizing the volume of an open-top box with a square base given a fixed surface area constraint.
Example showing how to minimize material cost for a rectangular container using a volume constraint and cost equation.
llustration and setup for finding the point on a line closest to a given external point using distance minimization.
Optimization example finding the rectangle of largest area that fits under a parabola with its base on the x-axis.
Example showing how to find the largest rectangle that can be inscribed in a right triangle using similar triangles and area maximization.
Example determining the dimensions of a poster with margins that maximize the printed area for a fixed total area.

4.8 & 4.9 - Newton’s Method and Antiderivatives

Example applying Newton’s Method to find the real root of the cubic equation x³ – 2x – 5 = 0 using iterative approximation.
Step-by-step example using Newton’s Method to approximate the square root of 2 through repeated iterations.
Example demonstrating how to use Newton’s Method to solve the equation cos x = x by successive approximations.
Step-by-step examples for finding the general antiderivative of various algebraic, exponential, and trigonometric functions.
Examples showing how to find the original function f(x) from its higher-order derivatives using integration and initial conditions.
Example of a motion problem where position is found by integrating acceleration and applying initial conditions to determine constants of integration.

Chapter 5 - Integrals

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5.1 - Area and Riemann Sums

Visual comparison of right-endpoint, left-endpoint, and midpoint rectangle methods for approximating the area under y = 16 − x² from x = 0 to 4.
Example showing how to write an expression for the area under a function as the limit of a Riemann sum over a given interval.
Example demonstrating how to identify the function and interval represented by a given limit in summation form to find the corresponding area.
Worked example demonstrating how to calculate a Riemann sum for f(x) = 2x – 4 using left and right endpoints over a given interval.

5.2 - The Definite Integral

Overview of definite integral notation, meaning, and basic properties, including limits of integration and additive intervals.
Example showing how to evaluate a definite integral by expressing it as the limit of a Riemann sum using summation formulas.
Examples of evaluating definite integrals by interpreting them as geometric areas under the curve, including absolute value and semicircular regions.
Graphical example illustrating how to compute accumulated area by evaluating g(x) = ∫ₐˣ f(t) dt for various x-values along a piecewise linear function.
Illustration of how to estimate the value of a definite integral using the inequality m(b – a) ≤ ∫ₐᵇ f(x) dx ≤ M(b – a).

5.3 - The Fundamental Theorem of Calculus

Examples demonstrating how to find the derivative of a function defined as a definite integral using the Fundamental Theorem of Calculus Part I.
ollection of practice problems featuring both indefinite and definite integrals with polynomial, trigonometric, and exponential functions.
Examples showing how to evaluate definite integrals using antiderivatives with the Fundamental Theorem of Calculus Part II.

5.4 & 5.5 - Indefinite Integrals and the Net Change Theorem

Example demonstrating the Net Change Theorem, showing how to compute total change in oil volume in a tank given a rate function over time.
Example problem showing how to find velocity and distance from an acceleration function using integration and initial conditions.
Examples demonstrating u-substitution for definite integrals, highlighting how to change limits of integration when substituting variables.
Examples showing how to evaluate indefinite integrals using u-substitution, including polynomial, exponential, and trigonometric functions.

Chapter 6 - Applications of Integration

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6.1 - Areas Between Curves

Example showing how to find the area between the line y = x + 2 and the parabola y = –x² + 4 using the area between curves formula.
Example illustrating how to find the area between the curves x = y² and x = y + 2 using integration with respect to y.
Example showing how to compute the area enclosed between the functions y = x² and y = √x by integrating with respect to x.
Example using symmetry to calculate the area enclosed between y = x³ and y = x.
Example demonstrating how to find the area between y = x³ – 2x² + 1 and y = x² – 2x + 1 by splitting the region into two parts.

6.2 & 6.3 - Volumes

Examples showing how to find the volume of solids of revolution by rotating regions about vertical lines such as x = 1 and x = –1 using the disk and washer methods.
Examples showing volume calculations for solids generated by rotating regions about horizontal lines such as y = 1 and y = –1 using the washer method.
Examples showing volume calculations for solids generated by rotating regions about horizontal lines such as y = 1 and y = –1 using the washer method.
Examples showing how to find the volume of solids of revolution by rotating regions about vertical lines such as x = 1 and x = –1 using the disk and washer methods.
Example showing how to compute the volume of a solid formed by rotating the curve y = 2x² − x³ about the y-axis using the shell method formula.
Example illustrating how to calculate the volume of a solid generated by rotating a region between y = 1 and y = 2 about the x-axis using cylindrical shells.
Example demonstrating how to find the volume of a solid of revolution formed by rotating a region about a horizontal line y = 3 using the shell method.

6.4 & 6.5 - Work and Average Value

Diagram illustrating Hooke’s Law with examples of work done by a spring during stretching and compression. Includes formulas W = ∫ F(x)dx and F(x) = kx, and notes on units in Newton·meters (Joules) and foot·pounds.
Examples showing how to find the average value of a function using the definite integral formula for f(x) = sin(x) on [0, π] and f(x) = 2x³ − 6x² on [−1, 2].

About Dr. Cheryl Hile

Understand The Math was created by Dr. Cheryl Hile, who earned a Ph.D. in Engineering Science and Applied Mathematics from Northwestern University. She has more than 25 years of experience teaching mathematics at the university level, including 20 years at Penn State. The lessons and guided notes on this site are based on materials she developed for her university courses.