Question 1

Evaluate the following limit using L'Hôpital's Rule:

$$
\lim_{x 	o 1} \frac{x^5$ - x^3$ - x^2$ + 1}{x^6$ - x^4$ - x^2$ + 1}
$$
Which of the following is the correct value?

Accepted Answer

$$\frac{1}{2}$$

Question 2

Compute the following limit using L'Hôpital's Rule:

$$
\lim_{x 	o \infty} \frac{x^2$ \ln x - x}{x^2$ + x \ln x}
$$
What is the value of this limit?

Accepted Answer

$$0$$

Question 3

Let $$g(x) = \sqrt{x} - \sqrt[3]{x}. $$What is $$g'(x)$$?

Accepted Answer

$$\frac{1}{2\sqrt{x}} - \frac{1}{3}x$$^{-2/3}$$

Question 4

Find the linear approximation $$L(x) to$$ $$g(x) = \sqrt{x} - \sqrt[3]{x} at$$ $$a = 64.$$

Accepted Answer

$$L(x) = 6 - 4 + \left(\frac{1}{16} - \frac{1}{48}\right)(x - 64)$$

Question 5

Using the linear approximation $$L(x)$$ found for $$g(x) = \sqrt{x} - \sqrt[3]{x} at$$ $$a = 64$$, estimate $$g(65).$$

Accepted Answer

$$2.0208$$

Question 6

Given $$g(x) = x^4$ + x - 1$$, and starting with $$x_0 = 1$, what is the value of x_1$ after one step of Newton's Method?

Accepted Answer

$$0.6$$

Question 7

Continuing Newton's Method for $$g(x) = x^4$ + x - 1$$ with $$x_0 = 1$ and x_1$ = 0.6$$, what is the value of $$x_2 ($$rounded to five decimal places)?

Accepted Answer

$$0.56794$$

Question 8

After three steps of Newton's Method for $$g(x) = x^4$ + x - 1$$ starting at $$x_0 = 1$, what is the value of x_3$ (rounded to five decimal places)?

Accepted Answer

$$0.56714$$

Question 9

Suppose you are modeling the cooling of a cup of coffee using Newton's Law of Cooling, which involves exponential functions. If the temperature $$T(t) at $$time $$t is $$given by $$T(t) = T_{env} + (T_0$ - T_{env})e$$^{-kt}$$, which calculus concept would you use to find the rate at which the coffee is cooling at any time t$?

Accepted Answer

The derivative $$\frac{dT}{dt}$$

Question 10

A pharmaceutical company wants to maximize the profit $$P(x)$$ from selling $$x$$ units of a drug, where $$P(x) is $$differentiable. Which calculus technique should they use to find the production level that yields maximum profit?

Accepted Answer

Set $$P'(x) = 0$$ and analyze the critical points

Question 11

Which of the following best describes the purpose of Newton's Method in calculus?

Accepted Answer

To approximate the roots of a real-valued function

Question 12

Suppose $$f(x) is $$continuous and differentiable, and $$f(a) = 0$$ but $$f'(a) 
eq 0. $$Which of the following statements is true about Newton's Method starting at $$x_0 = a$?

Accepted Answer

The next iterate $$x_1$$ will be $$x_1 = a$$ - \frac{f(a)}{f'(a)} = a$$

Question 13

A biologist is modeling the population $$P(t) of a $$bacteria culture, where $$P(t) = P_0$ e$$^{kt}$$. To estimate the population at t = 1.01$ given P(1$$)$$, which calculus technique would provide a quick approximation?

Accepted Answer

Linear approximation (tangent line) at $$t = 1$$

Question 14

Which of the following is a necessary condition for applying L'Hôpital's Rule to evaluate $$\lim_{x 	o a} \frac{f(x)}{g(x)}$$?

Accepted Answer

Both (a) and (b)

Question 15

Suppose $$h(x) = x^2$ \ln x - x. $$What is $$h'(x)$$?

Accepted Answer

$$2x \ln x + \ln x$$

Question 16

A chemical reaction's rate is modeled by $$R(t) = Ce$$^{-kt}$$. If you want to know how fast the reaction rate is changing at time t$, which calculus operation do you perform?

Accepted Answer

Compute $$R'(t)$$

Calculus Quiz 5: Limits, Derivatives, Linear Approximation, and Newton's Method

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