Calculus Skills Assessment

Rachel is standing atop a 13 ft ladder. The ladder is leaning against a vertical wall. The ladder starts sliding away from the wall at a rate of 3 ft/sec. How fast is the ladder sliding down the wall when the tip of the ladder is 5 ft high?

Evaluate the limit: $$\lim_{x \to 0} \frac{\sin(x)}{x}$$ .

What is the integral of $$\int 3x^2 dx$$ ?

Find the derivative of $$f(x) = \ln(x)$$ .

If a balloon is being filled with air such that its volume increases at a rate of $$5 cm^3/sec$$ , how fast is the radius of the balloon increasing when the radius is 2 cm? Assume the balloon is a perfect sphere.

What is the maximum value of the function $$f(x) = 3x^2 - 12x + 9$$ ?

Find the derivative of $$g(x) = e^{2x}$$ .

Evaluate the integral $$\int_0^1 (x^4 - 2x^2 + 1)dx$$ .

What is the limit of $$\frac{e^x - 1}{x}$$ as $$x$$ approaches 0?

If $$y = 3t^4 - 2t^3 + 7t - 5$$ , find $$\frac{dy}{dt}$$ .