Question

axis of revolution: $x$-axis, $v = \int_{a}^{b} \pi \left( f(x)^2 - g(x)^2 \
ight) dx$
axis of revolution: $y$-axis, $v = \int_{c}^{d} \pi \left( f(y)^2 - g(y)^2 \
ight) dy$
volume - shell method
axis of revolution: $x$-axis, $v = \int_{c}^{d} 2\pi y g(y) dy$
axis of revolution, $y$-axis, $v = \int_{a}^{b} 2\pi x f(x) dx$

question 15

evaluate: $\frac{d}{dx} \left x^4 - 5x^3 - 35x + 2 \
ight$

$\circ$ $4x^3 - 15x^2 - 35$

$\circ$ $5x^5 - 15x^4 - 35x^2 + 2x$

$\circ$ $(4x^3 - 15x^2)(35x + 2) + (x^4 - 5x^3)(35x + 2)$

$\circ$ no correct answer choice is given.

$\circ$ $x^4 - 5x^3 - 35x + 2 + c$

Explanation:

Step1: Apply Power Rule

The power rule for differentiation is $\frac{d}{dx}(x^n) = nx^{n - 1}$. For the term $x^4$, its derivative is $4x^{4 - 1}=4x^3$.

Step2: Differentiate $-5x^3$

Using the power rule, the derivative of $-5x^3$ is $-5\times3x^{3 - 1}=-15x^2$.

Step3: Differentiate $-35x$

The derivative of $-35x$ is $-35\times1x^{1 - 1}=-35$.

Step4: Differentiate the constant term

The derivative of a constant (2 in this case) is 0.

Step5: Combine the derivatives

Adding the derivatives of each term together: $4x^3-15x^2 - 35+0 = 4x^3-15x^2 - 35$.

Answer:

A. $4x^3 - 15x^2 - 35$