Calculus on Demand at Dartmouth College Lecture 14 | Index | Lecture 16 Lecture 15

## Resources

Math 3 Course Syllabus
Practice Exams

# Contents

In this lecture we take our first look at what it means to solve an equation involving derivatives. We also consider what we mean by the velocity and the acceleration of a moving object.

### Quick Question

Which is the graph of the velocity, and which the distance, for an object falling from rest near the surface of the earth?

### Outline

Outlines for
Antiderivatives and Initial Value Problems
Velocity and Acceleration

### Textbook

Antiderivatives and Initial Value Problems
Velocity and Acceleration

### Quiz

Antiderivatives and Initial Values Problems Quiz
Velocity and Acceleration Quiz

### Examples

• Graph sin2(x), then use the graph to sketch an antiderivative of sin2(x).
• Let g'''(x) = sin(x), with the conditions g(0) = 0, g'(0) = 1, g''(0) = −1. Find g(x).
• Find a solution to a 4th-order differential equation.
• A golfer on the moon (where gravitational acceleration equals 1.67 m/sec2) hits a ball whose initial velocity in the vertical direction is 30 meters per second. What is the maximum height the ball reaches?
• A meteoroid falling to Earth is discovered when it is at an altitude of 9000 kilometers, traveling at a velocity of 70 kilometers per second. Assuming acceleration due to Earth's gravity is constant, and neglecting air resistance, how fast will the meteoroid be falling when it hits the ground? What will its acceleration be?
• You drop a rock off Quechee Gorge Bridge and it hits the water below about 3.2 seconds later. Approximately how high is the bridge?

### Videos

• If f(x) is an antiderivative of f'(x), so is f(x) + constant
• y'(x) = cos(3x), y(0) = 5
• y'(x) = x12 + 1, y(1) = 15/13
• Ball is thrown upward 10 m/s initial velocity; how high does it go?
• Deriving s(t) = 16t2 + v0t + s0
• How long does it take Wile E. Coyote to fall off a 100-ft cliff?
• If Wile E. Coyote takes 20 seconds to reach the ground, how high is the cliff?
• If initial velocity is 3 ft/s upward, and he hits the ground with velocity 100 ft/s, how high is the cliff?

Lecture 14 | Index | Lecture 16