Differentiate implicitly the equation $$\sin(x)+\sin(y)=0$$ with respect to x and solve for $$\frac{dy}{dx}$$.

All of the following statements regarding linearization and differentials are true except:

If $$ f(x) = \sin\left(7x^2 + \frac{\pi}{4}\right) $$, then $$ f'\left(\sqrt{\frac{\pi}{28}}\right) = $$ ?

All of the following statements regarding the Constant Rule in differentiation are true except:

What is the area of the region enclosed by the graphs of

Given the function $$f(x)=x^3+x^2+x+1$$, evaluate f(x) at the x-values -2, -1, 0, and 1. The computed values are: f(-2) = -5, f(-1) = 0, f(0) = 1, and f(1) = 4. Rank the x-values in increasing order based on their corresponding f(x) values. Express your answer as an ordering separated by commas.

Let $$y = f(x)$$ be a differentiable function such that $$\displaystyle\frac{dy}{dx} = \frac{3x}{y}$$ and $$f(2) = 6$$. What is the approximation of $$f(2.2)$$ using the line tangent to the graph of $$f$$ at $$x = 2$$?

The continuous function $$ g $$ has domain $$ x \neq 3 $$ and satisfies $$ g(x) < -1 $$ for all $$ x $$ in its domain. If the graph of $$ g $$ has asymptotes $$ x = 3 $$ and $$ y = -1 $$, which of the following statements must be true?

A car's position is given by $$s(t)=4*t^2+2*t$$. Find the car's velocity at time t = 3 seconds.

Differentiate implicitly the equation $$\sin(x)+\cos(y)=0$$ with respect to x. Solve for $$\frac{dy}{dx}$$.

Let $$f$$ be the function given by $$\displaystyle f(x) = \int_0^x (3t^2 - 12t + 9) \, dt$$. What is the $$x$$-coordinate of the point of inflection of the graph of $$f$$?

Find the constant C such that the function $$F(x)=\frac{x^{3/2}}{2}-\frac{5*x^{1/2}}{2}+C$$ is an antiderivative of $$f(x)=\frac{3*x+2}{2*\sqrt{x}}$$ and satisfies $$F(4)=10.$$

Given the function $$f(t)=e^{t}+t*e^{t}$$, find its derivative at $$t=1$$.

A reservoir drains at a rate $$r(t) = 12 + 6\cos\bigl(\tfrac{\pi t}{12}\bigr)$$ gallons per hour, where $$t$$ is the number of hours from the start of draining. To the nearest gallon, what is the total amount of water drained from the reservoir from $$t = 0$$ to $$t = 24$$ hours?

For the function $$f(x)=2*x^5-3*x^3+ x^2-4*x+9$$, determine the derivative by applying the power rule term-by-term. Without using a calculator, select the equation that correctly represents $$f'(x)$$.

I. The limit $$\lim_{x\to0}\frac{\sin(3*x)}{x}$$ can be rewritten as $$3*\frac{\sin(3*x)}{3*x}$$. II. Since $$\lim_{x\to0}\frac{\sin(3*x)}{3*x} = 1$$, it follows that the limit is 3. III. Direct substitution of $$x=0$$ yields an indeterminate form (0/0), so the limit does not exist. Select the option that correctly identifies the valid statements for evaluating $$\lim_{x\to0}\frac{\sin(3*x)}{x}$$.

Evaluate the definite integral $$\int_{0}^{4} \sqrt{x}\,dx$$, rewriting $$\sqrt{x}$$ as $$x^{1/2}$$.

All of the following statements regarding the Fundamental Theorem of Calculus (FTC) are true except:

$$\displaystyle \int \left(5e^{2x} + \frac{1}{x}\right) dx =$$

All of the following statements concerning piecewise linear approximations of a function's graph are true except: