2.3 – Classwork (Basic Rules & Context)
Learning Objectives
By the end of this section, you should be able to:
- Apply derivative rules to solve mixed practice problems
- Connect derivative concepts to real-world applications
- Solve problems involving multiple derivative rules
- Interpret derivatives in various contexts
- Build confidence through extensive practice
Practice Problem Sets
Problem 1: Basic Differentiation
Find the derivative of \( f(x) = 4x^3 - 2x^2 + 7x - 5 \)
\[ f'(x) = \frac{d}{dx}[4x^3] - \frac{d}{dx}[2x^2] + \frac{d}{dx}[7x] - \frac{d}{dx}[5] \]
\[ f'(x) = 12x^2 - 4x + 7 \]
Problem 2: Motion Application
A ball is thrown upward with position function \( s(t) = -16t^2 + 64t + 80 \) feet. Find:
(a) The velocity function
(b) When the ball reaches its maximum height
(c) The velocity when the ball hits the ground
(a) Velocity function:
\[ v(t) = s'(t) = -32t + 64 \]
(b) Maximum height occurs when velocity is zero:
\[ -32t + 64 = 0 \]
\[ t = 2 \text{ seconds} \]
(c) Ball hits ground when \( s(t) = 0 \):
\[ -16t^2 + 64t + 80 = 0 \]
\[ t^2 - 4t - 5 = 0 \]
\[ (t-5)(t+1) = 0 \]
\[ t = 5 \text{ seconds} \]
\[ v(5) = -32(5) + 64 = -96 \text{ ft/s} \]
Problem 3: Related Rates
A ladder 10 feet long rests against a vertical wall. If the bottom of the ladder slides away from the wall at 1 ft/s, how fast is the top of the ladder sliding down the wall when the bottom is 6 feet from the wall?
Step 1: Set up the relationship
Let \( x \) = distance from wall to bottom of ladder
Let \( y \) = height of top of ladder
\[ x^2 + y^2 = 10^2 \]
Step 2: Differentiate with respect to time
\[ 2x\frac{dx}{dt} + 2y\frac{dy}{dt} = 0 \]
Step 3: Find values when \( x = 6 \)
\[ 6^2 + y^2 = 100 \]
\[ y^2 = 64 \]
\[ y = 8 \text{ ft} \]
Step 4: Solve for \( \frac{dy}{dt} \)
\[ 2(6)(1) + 2(8)\frac{dy}{dt} = 0 \]
\[ 12 + 16\frac{dy}{dt} = 0 \]
\[ \frac{dy}{dt} = -\frac{12}{16} = -0.75 \text{ ft/s} \]
The top is sliding down at 0.75 ft/s.
Quick Concept Check
Question 1: What is the derivative of \( f(x) = 5x^4 - 3x^2 + 2 \)?
Question 2: If \( s(t) \) represents position, what does \( s'(t) \) represent?