Calculus 1 Example

Problems

Step-by-step video solutions for limits, derivatives, integrals, and applications.

Learn Calculus I with Clear and Detailed Example Problems

Master Calculus 1 by practicing solved example problems! Each problem is broken down into step-by-step solutions so you can build confidence and prepare for quizzes, exams, and real-world applications. Perfect for brushing up on key skills, tackling tricky homework, or reviewing before a test.

How to Use This Page

Each example comes directly from our Calculus 1 Course. You can:

  • Watch the full lesson to learn the concept first.

  • Work through the example videos below for step-by-step practice.

Example Problems

Chapter 1 - Functions and Models (Review)

YouTube Playlist

Chapter 2 - Limits and Derivatives

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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.
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.
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.
Identify x values where f is discontinuous and explain why, then test continuity from the left or right
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.
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).
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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Differentiate f(x)=x^65, f(t)=1/t, g(x)=sqrt(x), and y=x^(-1/3) using the power rule.
Differentiate f(x)=2x^3-3x^2-4x and y=(x^2+4x+3)/sqrt(x). Find tangent and normal lines for y=2x^3-x^2+2 at (1,3).
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 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 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.
Evaluate limits as x→0: sin5x/3x, tan6x/sin2x, and (1−cosx)/sinx using fundamental trig limits.
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)
Identify inner and outer functions for y=sqrt(1+x^5), y=(2x^3+5)^4, and y=sin(tanx).
Chain rule formula: derivative of outer times derivative of inner. Differentiate y=(2x^3+5)^4 using both forms.
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)).
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).
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.
Analyze motion for s(t)=t^3−9t^2+15t+10: velocity, rest, direction, distance, acceleration, and speed changes.
Find the marginal cost when 200 items are produced for C(x)=5000+10x+0.01x^2.
Find the rate of population growth at t=4 hours for P(t)=5000/(1+9e^(-0.6t)).
Find the rate of change at t=10 minutes for concentration C(t)=0.05e^(-0.1t).
Find velocity, maximum height, and time to hit the ground for projectile h(t)=1.5+19.6t−4.9t^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.
Newton’s Law of Cooling: coffee cools from 200°F to 160°F in 15 minutes. Find temperature model, value at 30 minutes, and time to 100°F.
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 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 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 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.
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: 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 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: 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.
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.