Lecture 18

Q1. At 5 am the temperature is 42°F and increasing at a rate of 2°F per hour. Estimate the temperature at 5:15 am.

Background

Topic: Tangent Line Approximation (Linearization)

This question tests your ability to use the tangent line (linear) approximation to estimate the value of a function near a known point, given the function's value and rate of change (derivative) at that point.

Key Terms and Formulas

• Tangent Line Approximation: The tangent line to $y = f(x)$ at $x = a$ is $L(x) = f(a) + f'(a)(x - a)$.

• Derivative: $f'(a)$ is the rate of change of $f$ at $x = a$.

Step-by-Step Guidance

1. Let $f(x)$ be the temperature at time $x$ (in hours after midnight). At $x = 5$, $f(5) = 42$ and $f'(5) = 2$.

2. Write the tangent line approximation at $x = 5$ using the formula: $L(x) = f(5) + f'(5)(x - 5)$.

3. To estimate the temperature at 5:15 am, convert 15 minutes to hours: $5:15$ am is $x = 5.25$.

4. Plug $x = 5.25$ into the tangent line equation to set up the approximation for $f(5.25)$.

Try solving on your own before revealing the answer!

Q2. If $f(x)$ is the temperature at time $x$ am, find the equation of the tangent line to $y = f(x)$ at $x = 5$.

Background

Topic: Equation of the Tangent Line

This question asks you to write the equation of the tangent line to a function at a given point, using the function's value and derivative at that point.

Key Terms and Formulas

• Tangent Line Equation: $y = f(a) + f'(a)(x - a)$

Step-by-Step Guidance

1. Recall that at $x = 5$, $f(5) = 42$ and $f'(5) = 2$.

2. Substitute $a = 5$, $f(5)$, and $f'(5)$ into the tangent line formula.

3. Simplify the equation to express $y$ in terms of $x$.

Try solving on your own before revealing the answer!

Q3. Use this tangent line to predict the approximate temperature at 6 am.

Background

Topic: Tangent Line Approximation

This question asks you to use the tangent line equation from the previous part to estimate the function's value at a new point.

Key Terms and Formulas

• Tangent Line Approximation: $L(x) = f(a) + f'(a)(x - a)$

Step-by-Step Guidance

1. Recall the tangent line equation you found in the previous part.

2. To estimate the temperature at 6 am, set $x = 6$ in your tangent line equation.

3. Plug in the values and simplify the expression to set up the approximation for $f(6)$.

Try solving on your own before revealing the answer!

Q4. The tangent line approximation is used to estimate the temperature at the following times. Which do you think is most accurate? Why?

Background

Topic: Accuracy of Linear Approximation

This question tests your understanding of when the tangent line (linear) approximation is most accurate, based on how far the estimation point is from the point of tangency.

Key Concepts

• The tangent line approximation is most accurate near the point where the tangent is taken (here, $x = 5$).

• The further you move from $x = 5$, the less accurate the approximation becomes.

Step-by-Step Guidance

1. List the times given: 4 am, 4:50 am, 5:25 am, 6 am, midnight.

2. Determine which time is closest to 5 am (the point of tangency).

3. Explain why the approximation is more accurate for points closer to 5 am, and less accurate as you move further away.

Try explaining your reasoning before revealing the answer!

Q5. We are told that $f(4) = 2$ and $f'(4) = 3$. Find the equation of the tangent line to $y = f(x)$ at $x = 4$.

Background

Topic: Tangent Line Equation

This question asks you to write the equation of the tangent line to a function at a given point, using the function's value and derivative at that point.

Key Terms and Formulas

• Tangent Line Equation: $y = f(a) + f'(a)(x - a)$

Step-by-Step Guidance

1. Identify $a = 4$, $f(4) = 2$, and $f'(4) = 3$.

2. Substitute these values into the tangent line formula.

3. Simplify the equation to express $y$ in terms of $x$.

Try solving on your own before revealing the answer!

Q6. Use this tangent line approximation to estimate $f(4.1)$.

Background

Topic: Tangent Line Approximation

This question asks you to use the tangent line equation to estimate the value of the function at a nearby point.

Key Terms and Formulas

• Tangent Line Approximation: $L(x) = f(a) + f'(a)(x - a)$

Step-by-Step Guidance

1. Recall the tangent line equation you found in the previous part.

2. Set $x = 4.1$ in your tangent line equation.

3. Plug in the values and simplify the expression to set up the approximation for $f(4.1)$.

Try solving on your own before revealing the answer!

Q7. Use the tangent line approximation to estimate the value of $x$ which gives $f(x) = 2.9$.

Background

Topic: Solving for $x$ Using Linear Approximation

This question asks you to use the tangent line equation to estimate the value of $x$ for which the function reaches a certain value.

Key Terms and Formulas

• Tangent Line Equation: $y = f(a) + f'(a)(x - a)$

Step-by-Step Guidance

1. Set $y = 2.9$ in your tangent line equation from earlier.

2. Solve the equation for $x$ to estimate the value where $f(x) = 2.9$.

3. Isolate $x$ and simplify the expression, stopping before the final calculation.

Try solving on your own before revealing the answer!

Q8. If $f(t)$ is the number of grams of a chemical reagent after $t$ seconds, $f(0) = 20$, and $f'(0) = -3$, approximately how many grams are there after $t$ seconds?

Background

Topic: Linear Approximation (Tangent Line)

This question asks you to use the tangent line approximation to estimate the value of a function at a small time $t$, given its initial value and rate of change.

Key Terms and Formulas

• Tangent Line Approximation: $L(t) = f(0) + f'(0)t$

Step-by-Step Guidance

1. Identify $f(0) = 20$ and $f'(0) = -3$.

2. Write the tangent line approximation formula for $f(t)$ at $t = 0$.

3. Plug in the values to set up the linear approximation for $f(t)$.

Try solving on your own before revealing the answer!

Q9. Let $g(x) = \sqrt{x}$. We are told that $g'(25) = \frac{1}{10}$ and $\sqrt{25} = 5$. Use the tangent line approximation at $x = 25$ to get an approximate value for $\sqrt{26}$.

Background

Topic: Tangent Line Approximation for Square Roots

This question asks you to use the tangent line approximation to estimate the square root of a number near a known value.

Key Terms and Formulas

• Tangent Line Approximation: $L(x) = g(25) + g'(25)(x - 25)$

Step-by-Step Guidance

1. Identify $g(25) = 5$ and $g'(25) = \frac{1}{10}$.

2. Write the tangent line equation at $x = 25$.

3. Set $x = 26$ in your tangent line equation to set up the approximation for $\sqrt{26}$.

Try solving on your own before revealing the answer!

Q10. A calculator tells you that $\sqrt{26} = 5.09902...$. Find the error and percentage error in the tangent line approximation.

Background

Topic: Error Analysis in Linear Approximation

This question asks you to compare your tangent line approximation to the actual value and compute the error and percentage error.

Key Terms and Formulas

• Error: $\text{Error} = |\text{Approximate Value} - \text{Actual Value}|$

• Percentage Error: $\text{Percentage Error} = \frac{\text{Error}}{\text{Actual Value}} \times 100\%$

Step-by-Step Guidance

1. Recall your tangent line approximation for $\sqrt{26}$ from the previous part.

2. Subtract the actual value ($5.09902...$) from your approximation to find the error.

3. Divide the error by the actual value and multiply by 100 to set up the percentage error calculation.

Try solving on your own before revealing the answer!

Q11. If $f(t)$ is the capacity in acre-feet of Lake Cachuma at $t$ years after 1950, and you know the capacity in 1950 and 2010, what is the linear approximation for $f(t)$?

Background

Topic: Linear Approximation from Data

This question asks you to use two data points to find a linear approximation (tangent line) for a function over time.

Key Terms and Formulas

• Slope (Rate of Change): $m = \frac{f(t_2) - f(t_1)}{t_2 - t_1}$

• Linear Approximation: $L(t) = f(t_1) + m(t - t_1)$

Step-by-Step Guidance

1. Identify the two data points: $(t_1, f(t_1)) = (0, 205{,}000)$ and $(t_2, f(t_2)) = (60, 190{,}000)$, where $t$ is years after 1950.

2. Calculate the slope $m$ using the two points.

3. Write the linear approximation formula using $f(0)$ and the slope $m$.

Try solving on your own before revealing the answer!

Q12. Estimate the year in which 10% of the original capacity of Lake Cachuma will have been lost due to silt.

Background

Topic: Applying Linear Approximation to Predict Future Values

This question asks you to use your linear approximation to estimate when the capacity will have decreased by 10% from its original value.

Key Terms and Formulas

• Original Capacity: $205{,}000$ acre-feet

• 10% Loss: $0.10 \times 205{,}000 = 20{,}500$ acre-feet lost

• Target Capacity: $205{,}000 - 20{,}500 = 184{,}500$ acre-feet

Step-by-Step Guidance

1. Set $f(t) = 184{,}500$ in your linear approximation equation from the previous part.

2. Solve for $t$ to find the number of years after 1950 when this capacity is reached.

3. Add $t$ to 1950 to estimate the year.

Try solving on your own before revealing the answer!