Calculus for Healthcare Professionals: Master Limits, Derivatives, and Integrals

Question 1

Find the derivative of the function f(x) = x^3 + 2x^2 - 5x + 1.

Accepted Answer

3x^2 + 4x - 5

Answer

x^3 + 4x^2 - 5

Answer

3x^2 - 4x + 1

Answer

x^3 - 2x^2 + 5x - 1

Question 2

Calculate the integral of the function f(x) = sin(x) from 0 to π/2.

Accepted Answer

1

Answer

0

Answer

-1

Answer

π/2

Question 3

Which of the following formulas represents the area under the curve y = f(x) from a to b?

Accepted Answer

∫[a, b] f(x) dx

Answer

∫[b, a] f(x) dx

Answer

∫[0, a] f(x) dx

Answer

∫[0, b] f(x) dx

Question 4

Calculate the derivative of the function y = ln(x^2 + 1).

Accepted Answer

2x / (x^2 + 1)

Answer

x / (x^2 + 1)

Answer

(x^2 + 1) / x

Answer

x^2 / (x^2 + 1)

Question 5

Determine the convergence or divergence of the infinite series ∑[n=1 to ∞] 1 / n.

Accepted Answer

Diverges

Answer

Converges

Answer

Alternates

Answer

Conditional

Question 6

Find the limit of the function (x^2 - 1) / (x - 1) as x approaches 1.

Accepted Answer

1

Answer

0

Answer

2

Answer

The limit does not exist.

Question 7

Calculate the derivative of the function f(x) = 3x^2 + 2x - 5.

Accepted Answer

6x + 2

Answer

3x^2 - 2x + 5

Answer

9x^2 + 2

Answer

3x - 5

Question 8

Find the derivative of the function f(x) = e^(2x + 1).

Accepted Answer

2e^(2x + 1)

Answer

e^(2x + 1)

Answer

2e^x

Answer

e^x

Question 9

Which integration technique should be used to find the integral of the function f(x) = x^2 + 1?

Accepted Answer

Substitution

Answer

Integration by parts

Answer

Trigonometric substitution

Answer

Partial fractions

Question 10

A patient is given a dose of 500mg of a drug. The amount of drug remaining in the patient's body after t hours is given by the function f(t) = 500e^(-0.05t). What is the half-life of the drug?

Accepted Answer

13.86 hours

Answer

27.72 hours

Answer

55.44 hours

Answer

110.88 hours

Question 11

Determine the area under the curve y = x^2 between x = 1 and x = 3.

Accepted Answer

8

Answer

4

Answer

12

Answer

16

Question 12

Calculate the indefinite integral of f(x) = 3x^2 + 2x - 1 with respect to x.

Accepted Answer

3x^3 + 2x^2 - x + C

Answer

x^3 + x^2 - x + C

Answer

3x^3 + x^2 - x + C

Answer

x^3 + 2x^2 - x + C

Question 13

What is the derivative of the logarithmic function f(x) = log2(x)?

Accepted Answer

1 / (x * ln2)

Answer

ln2 / x

Answer

x * ln2

Answer

1 / x

Question 14

Find the indefinite integral of the exponential function f(x) = e^(2x) with respect to x.

Accepted Answer

(1 / 2) * e^(2x) + C

Answer

e^(2x) + C

Answer

2 * e^(2x) + C

Answer

(1 / 2) * e^(x) + C

Question 15

Find the limit of (1 + 1 / x)^x as x approaches infinity.

Accepted Answer

e

Answer

0

Answer

1

Answer

infinity

Question 16

What is the derivative of the function f(x) = |x|?

Accepted Answer

1 if x > 0, 0 if x = 0, -1 if x < 0

Answer

1 if x > 0, -1 if x < 0

Answer

0

Answer

x

Question 17

Which of the following is the equation of the normal line to the parabola y = x^2 at the point (1, 1)?

Accepted Answer

y = -x + 2

Answer

y = x - 2

Answer

y = x + 2

Answer

y = -x - 2

Question 18

Calculate the derivative of the function f(x) = e^(2x + 1).

Accepted Answer

2e^(2x + 1)

Answer

e^(2x)

Answer

2e^x

Answer

e^2x

Question 19

Calculate the area under the curve y = sin(x) from 0 to π/2.

Accepted Answer

1

Answer

0

Answer

2

Question 20

Find the antiderivative of the function f(x) = 1/x.

Accepted Answer

ln|x| + C

Answer

1/x^2 + C

Answer

x + C

Answer

x^2 + C

Question 21

Which of the following accurately defines the limit of a function?

Accepted Answer

As the input value approaches a certain value, the function's output approaches a specific number.

Answer

The rate at which the function changes.

Answer

The precise value of the function at a particular input.

Question 22

Find the derivative of the function f(x) = x^2.

Accepted Answer

2x

Answer

x

Answer

x^3

Answer

2

Question 23

Which of the following demonstrates a practical application of calculus in the field of healthcare?

Accepted Answer

Determining appropriate drug dosages.

Answer

Maintaining patient records.

Answer

Diagnosing specific illnesses.

Answer

Administering patient treatment.

Question 24

Find the derivative of the function f(x) = e^x.

Accepted Answer

e^x

Answer

e^x / x

Answer

x * e^x

Question 25

Identify an example of a healthcare problem that can be addressed using calculus.

Accepted Answer

Calculating the rate of drug absorption in the body.

Answer

Diagnosing an illness based on a blood sample.

Answer

Administering a new medication to a patient.

Question 26

Find the derivative of the function f(x) = x² + 2x.

Accepted Answer

2x + 2

Answer

2

Answer

x³

Answer

x

Question 27

What is the derivative of f(x) = sin(x)?

Accepted Answer

cos(x)

Answer

sin(x)cos(x)

Answer

-sin(x)

Answer

xcos(x)

Question 28

Calculate the value of the definite integral ∫[0,1] (x² + 1) dx.

Accepted Answer

1.5

Answer

2.5

Answer

0.5

Answer

3.5

Question 29

Find the derivative of the function f(x) = e^x.

Accepted Answer

e^x

Answer

x * e^x

Answer

e^(x²)

Answer

1 / e^x

Question 30

Which of the following is NOT a step involved in finding the derivative of a function using the limit definition?

Accepted Answer

Graphing the function

Answer

Factoring out the limit

Answer

Simplifying the expression within the limit

Answer

Taking the limit of the difference quotient

Question 31

Find the derivative of x^3 using the power rule.

Accepted Answer

3x^2

Answer

x^2

Answer

x^4

Answer

1/x

Question 32

Which of the following techniques is used to determine the total area under a curve between two points on the x-axis?

Accepted Answer

Integration

Answer

Factoring

Answer

Differentiation

Answer

Substitution

Question 33

State the chain rule for derivatives.

Accepted Answer

If f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x)

Answer

If f(x) = g(h(x)), then f'(x) = g(h'(x)) * h'(x)

Answer

If f(x) = g(h(x)), then f'(x) = h'(x) / g'(x)

Answer

If f(x) = g(h(x)), then f'(x) = g'(x) / h'(x)

Question 34

Find the derivative of e^x using the chain rule.

Accepted Answer

e^x

Answer

-e^x

Answer

x * e^x

Answer

1 / e^x

Question 35

Which of the following is the indefinite integral of the function f(x) = sin(x)?

Accepted Answer

-cos(x) + C

Answer

-sin(x) + C

Answer

cos(x) + C

Answer

sin(x) + C

Question 36

What is the second derivative of the function f(x) = e^x?

Accepted Answer

e^x

Answer

e^(2x)

Answer

0

Answer

2e^x

Question 37

What is the limit as x approaches infinity of the function f(x) = (x^2 + 1)/(x - 1)?

Accepted Answer

Infinity

Answer

-1

Answer

0

Answer

1