Business Calculus Practice: Integrals, Tangent Lines, and Applications

Q1. Evaluate the integral:

$\int_{-5}^{-8} \frac{12x^5 - x^3 + 10x^7 - 3 - e^x + 1}{6x} \, dx$

Background

Topic: Definite Integrals & Integration Techniques

This question tests your ability to evaluate a definite integral involving a rational function and exponential terms. It requires knowledge of basic integration rules and properties.

Key Terms and Formulas:

• Definite Integral: $\int_a^b f(x) \, dx$

• Power Rule for Integration: $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$

• Exponential Rule: $\int e^x \, dx = e^x + C$

• Sum/Difference Rule: $\int [f(x) + g(x)] \, dx = \int f(x) \, dx + \int g(x) \, dx$

Step-by-Step Guidance

1. Rewrite the integrand by separating each term in the numerator over the denominator $6x$.

2. Express each term as a simpler function, for example $\frac{12x^5}{6x} = 2x^4$.

3. Integrate each term individually using the appropriate rules (power rule, exponential rule, etc.).

4. Set up the definite integral by plugging in the upper and lower limits for each integrated term.

Try solving on your own before revealing the answer!

Q2. Evaluate the integral:

$\int_{-4}^{5} \frac{2x^5 + 5 - 2\sqrt{x} + e^x + 6x}{x} \, dx$

Background

Topic: Definite Integrals & Integration Techniques

This question tests your ability to integrate a function with polynomial, radical, and exponential terms divided by $x$.

Key Terms and Formulas:

• Logarithmic Rule: $\int \frac{1}{x} \, dx = \ln|x| + C$

• Power Rule: $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$

• Exponential Rule: $\int e^x \, dx = e^x + C$

Step-by-Step Guidance

1. Break up the integrand so each term is divided by $x$.

2. Rewrite $\frac{2x^5}{x}$ as $2x^4$, $\frac{5}{x}$ as $5\frac{1}{x}$, $\frac{-2\sqrt{x}}{x}$ as $-2x^{-1/2}$, etc.

3. Integrate each term using the appropriate rule.

4. Set up the definite integral by substituting the upper and lower bounds.

Try solving on your own before revealing the answer!

Q3. The slope of a tangent line to a curve is given by $f'(x) = \frac{4x-20}{x^2 - 10x + 26}$. If the point (5, 6) is on the curve, find an equation of the curve.

Background

Topic: Antiderivatives & Initial Conditions

This question tests your ability to find the original function given its derivative and a point on the curve (initial condition).

Key Terms and Formulas:

• Antiderivative: $f(x) = \int f'(x) \, dx + C$

• Initial Condition: Use the given point to solve for $C$.

Step-by-Step Guidance

1. Set up the integral: $f(x) = \int \frac{4x-20}{x^2 - 10x + 26} \, dx + C$.

2. Consider substitution: Let $u = x^2 - 10x + 26$, then $du = 2x - 10 \, dx$; adjust as needed for the numerator.

3. Integrate using substitution or partial fractions if necessary.

4. Use the point (5, 6) to solve for the constant $C$.

Try solving on your own before revealing the answer!

Q4. The slope of a tangent line to a curve is given by $f'(x) = -6x^2 + 6x - 6$. If the point (0, 6) is on the curve, find the equation of the curve.

Background

Topic: Antiderivatives & Initial Conditions

This question tests your ability to find the original function from its derivative and a given point.

Key Terms and Formulas:

• Power Rule for Antiderivatives: $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$

• Initial Condition: Use the given point to solve for $C$.

Step-by-Step Guidance

1. Integrate $f'(x)$ term by term to find $f(x)$.

2. Apply the power rule to each term.

3. Add the constant of integration $C$.

4. Use the point (0, 6) to solve for $C$.

Try solving on your own before revealing the answer!

Q5. The rate of growth of profit (in millions of dollars) from a new technology is approximated by $P'(x) = x e^{-x^2}$ where $x$ represents time measured in years. The profit in the third year is $10,000. Find the profit function.

Background

Topic: Applications of Integration (Business Context)

This question tests your ability to find a profit function from its rate of change, using integration and an initial value.

Key Terms and Formulas:

• Antiderivative: $P(x) = \int P'(x) \, dx + C$

• Substitution for integrals involving $x e^{-x^2}$

• Initial Condition: Use the value at $x=3$ to solve for $C$.

Step-by-Step Guidance

1. Set up the integral: $P(x) = \int x e^{-x^2} \, dx + C$.

2. Use substitution: Let $u = -x^2$, then $du = -2x \, dx$.

3. Rewrite the integral in terms of $u$ and integrate.

4. Use the given profit at $x=3$ to solve for $C$.

Try solving on your own before revealing the answer!

Q6. A ball is rolled up a hill with initial velocity of 4 meters per second. If the acceleration of the ball is given by $a(t) = -3t + 6$ where $t$ is in seconds, find how far the ball is from where it started after 3 seconds.

Background

Topic: Applications of Integration (Motion)

This question tests your ability to use integration to find displacement from acceleration, given initial velocity.

Key Terms and Formulas:

• Velocity: $v(t) = \int a(t) \, dt + v_0$

• Displacement: $s(t) = \int v(t) \, dt + s_0$

• Initial velocity $v_0 = 4$ m/s

Step-by-Step Guidance

1. Integrate $a(t)$ to find $v(t)$, including the initial velocity.

2. Integrate $v(t)$ to find $s(t)$, the position function.

3. Evaluate $s(3)$ to find the displacement after 3 seconds.

4. Subtract $s(0)$ to find the distance from the starting point.

Try solving on your own before revealing the answer!