Question 1

Suppose the temperature at 5 am is $$42^{\circ} F$ and is increasing at a rate of 2$$^{\circ}$$ F$$ per hour. Using the tangent line approximation, estimate the temperature at 5:15 am.

Accepted Answer

$$42.5^{\circ} F$

Question 2

Given $$f(x) is $$the temperature at time $$x am$$, $$f(5) = 42$$, and $$f'(5) = 2. $$What is the equation of the tangent line to $$y = f(x) at$$ $$x = 5$$?

Accepted Answer

$$y = 2(x - 5) + 42$$

Question 3

Using the tangent line approximation from the previous question, estimate the temperature at 6 am.

Accepted Answer

$$44^{\circ} F$

Question 4

The tangent line approximation is most accurate when estimating the value of a function at which of the following times, given $$f(5)$$ and $$f'(5)$$?

Accepted Answer

At $$x = 5 (i.e$$., 5 am)

Question 5

Given $$f(4) = 2$$ and $$f'(4) = 3$$, what is the equation of the tangent line to $$y = f(x) at$$ $$x = 4$$?

Accepted Answer

$$y = 3(x - 4) + 2$$

Question 6

Using the tangent line from the previous question, estimate $$f(4.1).$$

Accepted Answer

$$2.3$$

Question 7

Suppose $$f(4) = 2$$ and $$f'(4) = 3. $$Using the tangent line approximation, estimate the value of $$x$$ for which $$f(x) = 2.9.$$

Accepted Answer

$$x = 4.3$$

Question 8

If $$f(t) is $$the number of grams of a chemical reagent after $$t$$ seconds, $$f(0) = 20$$, and $$f'(0) = -3$$, use the tangent line approximation to estimate the amount of reagent after $$t = 2$$ seconds.

Accepted Answer

$$17$$ grams

Question 9

Let $$g(x) = \sqrt{x}$$, $$g'(25) = \frac{1}{10}$$, and $$\sqrt{25} = 5. $$Use the tangent line approximation at $$x = 25 to $$estimate $$\sqrt{26}.$$

Accepted Answer

$$5.1$$

Question 10

A calculator gives $$\sqrt{26} \approx 5.09902. $$Using the tangent line approximation $$5.1$$, what is the absolute error and percentage error?

Accepted Answer

Absolute error: $$0.00098$$, Percentage error: $$0.0192\%$$

Question 11

Lake Cachuma originally had a capacity of $$205,000$$ acre-feet in 1950 and $$190,000$$ acre-feet in 2010. Let $$f(t) be $$the capacity $$t$$ years after 1950. What is the linear approximation for $$f(t)$$?

Accepted Answer

$$f(t) \approx 205,000 - 250t$$

Question 12

Using the linear approximation for Lake Cachuma, estimate the year in which $$10\% of $$the original capacity will have been lost due to silt.

Accepted Answer

2032

Question 13

Which of the following best describes why the tangent line approximation is most accurate near the point of tangency?

Accepted Answer

Because the function and its tangent line have the same value and slope at the point of tangency.

Question 14

Suppose $$f(x) is $$differentiable and $$f(a) = b$$, $$f'(a) = m. $$Which formula gives the tangent line approximation to $$f(x)$$ near $$x = a$$?

Accepted Answer

$$f(x) \approx m(x - a) + b$$

Question 15

A medication's concentration in the bloodstream is modeled by $$C(t)$$, with $$C(2) = 10 mg/L $$and $$C'(2) = -1.5 mg/L $$per hour. Estimate the concentration at $$t = 2.5$$ hours.

Accepted Answer

$$9.25 mg/L$$

Question 16

If $$h(x) is a $$function with $$h(3) = 7$$ and $$h'(3) = -2$$, use the tangent line approximation to estimate $$h(2.8).$$

Accepted Answer

$$7.4$$

Lecture 18

Study Guide - Practice Questions

Study Guide - Flashcards