Question 1

A pharmaceutical company is modeling the concentration of a drug in the bloodstream by the function C(t) = $$2t^2 - 3t + 4$$, where t is time in hours. What is the slope of the tangent line to the concentration curve at t = 2 hours, and what does this slope represent in this context?

Accepted Answer

The slope is 5, representing the rate at which the drug concentration is increasing at t = 2 hours.

Question 2

Which of the following best describes a tangent line to the curve y = f(x) at x = a?

Accepted Answer

A line that passes through the point (a, f(a)) and has the same slope as the curve at that point.

Question 3

Evaluate the following limit using appropriate algebraic techniques: lim_{x→1} $$(x^3$$ - $$1)/(x^4 - 1).$$

Accepted Answer

1/4

Question 4

A biologist is studying the growth of a bacterial colony modeled by f(x) = $$x^3. $$Using the definition of the derivative, what is the general formula for the slope of the tangent line at any point x?

Accepted Answer

$$3x^2$$

Question 5

For the function y = 10 - $$x^2$$, at which x-value does the tangent line become horizontal, and what is the significance of this point?

Accepted Answer

x = 0; this is the maximum point of the parabola.

Question 6

A patient’s heart rate is modeled by H(t) = 4t - 5. What is the equation of the tangent line to H(t) at t = 2 minutes?

Accepted Answer

y = 3 + 4(t - 2)

Question 7

Evaluate the following limit: lim_{x→0} (sin 5x)/x.

Accepted Answer

5

Question 8

Which of the following statements about limits is correct?

Accepted Answer

If lim_{x→a} f(x) = L, then as x approaches a, f(x) approaches L.

Question 9

A chemical reaction’s rate is modeled by R(x) = 1/x. What is the slope of the tangent line at x = 2?

Accepted Answer

-1/4

Question 10

Evaluate the limit: lim_{x→-2} $$((x+2)^2$)/(x^2 - 16).$$

Accepted Answer

0

Question 11

A function f(x) has a derivative f'(x) = 0 at x = a. What does this imply about the tangent line at x = a?

Accepted Answer

The tangent line is horizontal at x = a.

Question 12

Which of the following is an indeterminate form that requires algebraic manipulation to evaluate the limit?

Accepted Answer

lim_{x→a} (0/0)

Question 13

A medical researcher is analyzing the function f(x) = $$x^3 to $$predict the rate of change at x = 2. Using the difference quotient, what is the value of the slope-predictor at x = 2?

Accepted Answer

12

Question 14

Evaluate the limit: lim_{t→-2} $$(t^2$$ + $$2)/(t^2 + 4).$$

Accepted Answer

3/4

Question 15

Which of the following is a correct application of the key trigonometric limit lim_{x→0} (sin x)/x?

Accepted Answer

lim_{x→0} (sin 3x)/x = 3

Question 16

A parabola y = $$x^2$$ passes through the point (3, 0). Besides the x-axis, what is the equation of the other line through (3, 0) that also passes through another point on the parabola?

Accepted Answer

y = -6x + 18

Question 17

Evaluate the limit: lim_{x→-3} $$(3x^2 + 7x - 12).$$

Accepted Answer

-6

Question 18

Which of the following best explains why the limit lim_{x→0} $$(sin^2$$ $$x)/(x^2) $$equals 1?

Accepted Answer

Because (sin x)/x approaches 1 as x approaches 0, so (sin $$x/x)^2$$ also approaches 1.

Chapter 2: Prelude to Calculus – Tangent Lines, Slope Predictors, and Limits

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