Section I Part B — Full Practice Exam 2

Questions in this quiz (15)

1. Let $$f$$ be a function such that the rate of change of the derivative of $$f$$ is constant and equal to $$3$$ for all $$x$$. If $$f'(1)=2$$ and $$f(1)=4$$, find $$f(3)$$.

   • $$10$$
   • $$12$$
   • $$13$$
   • $$14$$

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

   • $$\frac{1}{2a}$$
   • $$\frac{1}{2a}$$
   • $$\frac{1}{3a}$$
   • $$0$$

3. Population $$y$$ grows according to the equation $$\frac{dy}{dt}=ky$$ where $$k$$ is a constant and $$t$$ is measured in years. If the population doubles every $$4$$ years, then $$k=$$

   • $$0.173$$
   • $$0.250$$
   • $$0.346$$
   • $$0.693$$

4. The radius of a circle is increasing at a rate of $$3$$ inches per minute. When the radius is $$5$$ inches, how fast is the area increasing in square inches per minute?

   • $$15\pi$$
   • $$20\pi$$
   • $$30\pi$$
   • $$45\pi$$

5. Let $$f(x)=\int_0^{3x}e^t\,dt$$. What value on $$[0,2]$$ satisfies the Mean Value Theorem for $$f$$?

   • $$1.210$$
   • $$1.402$$
   • $$1.732$$
   • $$1.910$$

6. The position for a particle is $$s(t)=-2t^3+3t^2+1$$. At what time $$t$$ on $$[0,2]$$ is the instantaneous velocity equal to the average velocity?

   • $$0.422$$
   • $$0.667$$
   • $$1.215$$
   • $$1.264$$

7. Let $$f(x)=x^2+2x+1$$. Using the trapezoidal rule with $$n=4$$, approximate $$\int_0^2 f(x)\,dx$$.

   • $$8.25$$
   • $$8.50$$
   • $$8.75$$
   • $$9.00$$

8. Let $$f(x)=4\ln(3x)$$ and $$g(x)=x^3+x$$. At what value of $$x$$ do the graphs have parallel tangent lines?

   • $$0.412$$
   • $$0.693$$
   • $$1$$
   • $$0.882$$

9. The derivative of a function is $$f'(x)=\frac{e^{-x}}{x}-\cos x$$. How many critical values does $$f$$ have on $$(0,8)$$?

   • One
   • Two
   • Three
   • Four

10. Evaluate $$\frac{d}{dx}\left(\int_x^4 f'(t)\,dt\right)$$

    • $$f'(x)$$
    • $$-f'(x)$$
    • $$f'(4)$$
    • $$-f'(4)$$

11. Let$$f(x)=\sqrt{x}+k$$, $$x$$<$$1$$.$$f(x)=ln(x+1)$$, $$x\ge1$$For what value of $$k$$ is $$f$$ continuous at $$x=1$$?

    • $$-1$$
    • $$0$$
    • $$\ln2-1$$
    • $$1$$

12. 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
    • I and II
    • I and III

13. The graph of the derivative of $$f$$ is shown above. Which of the following statements is true?

    • $$f(0)<f(6)<f(2)<f(4)$$
    • $$f(6)<f(0)<f(2)<f(4)$$
    • $$f(0)<f(2)<f(4)<f(6)$$
    • $$f(2)

14. The graph of $$f$$ is shown above. If $$g(x)=\int_a^x f(t)\,dt$$, for what value of $$x$$ does $$g(x)$$ have a relative minimum?

    • $$a$$
    • $$b$$
    • $$c$$
    • $$d$$

15. The graph of a function $$f$$ is shown above. Which of these statements about $$f$$ is false?

    • $$f$$ is continuous but not differentiable at $$x=a$$.
    • The left-hand and right-hand derivatives at $$x=a$$ are not equal.
    • $$f(a)$$ is defined, but $$f(c)$$ is not.
    • $$\lim_{x\to c^-}f(x)\ne\lim_{x\to c^+}f(x)$$