Section I Part B — Full Practice Exam 5

Questions in this quiz (12)

1. Let $$f$$ be the function given by $$f(x)=2\ln(3x)$$ and let $$g$$ be the function given by $$g(x)=x^3+3x$$. At what value of $$x$$ do the graphs of $$f$$ and $$g$$ have parallel tangent lines?

   • $$-0.612$$
   • $$0.412$$
   • $$0.612$$
   • $$1.000$$

2. Let $$f$$ be some function such that the rate of increase of the derivative of $$f$$ is $$3$$ for all $$x$$. If $$f'(1)=2$$ and $$f(0)=1$$, find $$f(2)$$.

   • $$3$$
   • $$4$$
   • $$5$$
   • $$6$$

3. Evaluate $$\lim_{x\to a}\frac{x-a}{x^2-a^2}$$.

   • $$\frac{1}{2a}$$
   • $$\frac{1}{a}$$
   • $$0$$
   • It does not exist

4. Let $$f(x)=\int_0^{3x}e^t\,dt$$. What value on $$[0,3]$$ satisfies the mean value theorem for $$f$$?

   • $$1.245$$
   • $$1.386$$
   • $$1.732$$
   • $$2.079$$

5. The position for a particle moving on the $$x$$-axis is given by $$s(t)=-t^3+3t^2+1$$. At what time $$t$$, on $$[0,2]$$, is the particle's instantaneous velocity equal to its average velocity over $$[0,2]$$?

   • $$0.423$$
   • $$0.667$$
   • $$1.155$$
   • $$1.577$$

6. Let $$f(x)=x^2+2$$. Using the trapezoidal rule, with $$n=4$$, approximate $$\int_0^4 f(x)\,dx$$.

   • $$26.00$$
   • $$28.00$$
   • $$29.00$$
   • $$30.00$$

7. Population $$y$$ grows according to the equation $$\frac{dy}{dt}=ky$$. If the population doubles every four years, then $$k=$$.

   • $$0.139$$
   • $$0.173$$
   • $$0.220$$
   • $$0.346$$

8. The radius of a circle is increasing at a rate of $$\frac{1}{2}$$ inch per minute. When the radius is $$4$$ inches, how fast is the area increasing in square inches per minute?

   • $$2\pi$$
   • $$3\pi$$
   • $$4\pi$$
   • $$5\pi$$

9. Let $$f(x)=x^2+2x-3$$. The tangent line to the graph at $$x=1$$ is used to approximate values of $$f$$. For what value(s) of $$x$$ is the tangent line approximation twice that of $$f$$? I. $$-1$$ II. $$0$$ III. $$2$$

   • I only
   • II only
   • III only
   • I and II

10. The first derivative of a function $$f$$ is given by $$f'(x)=\cos x-\frac{1}{x}$$. How many critical values does $$f$$ have on the open interval $$(1,10)$$?

    • One
    • Two
    • Three
    • Four

11. Evaluate $$\frac{d}{dx}\left(\int_x^{3x}f(t)\,dt\right)$$

    • $$3f(3x)-f(x)$$
    • $$f(3x)-f(x)$$
    • $$3f(3x)+f(x)$$
    • $$3f(3x)-3f(x)$$

12. Let $$f$$ be defined as $$f(x)=\begin{cases}x^2+k,&x<2\\\ln(x),&x\ge2\end{cases}$$ For a constant $$k$$, for what value of $$k$$ will $$\lim_{x\to2^-}f(x)=\lim_{x\to2^+}f(x)$$

    • $$-3$$
    • $$-2$$
    • $$-1$$
    • $$0$$