1. Let $$f(x)=|x+1|$$. Which of the following statements about f are true?
I.
f is continuous at x = -1
II.
f is differentiable at x = -1
III.
f has a cusp at x = -1
I only
II only
III only
*I and III only
I and II only
2. The normal line is the line that is perpendicular to the tangent line at a given x
value.
If the line normal to the graph of f at the point (1,2) passes through the point (1,1), then which of the following gives the value of f’(1)?
*$$-2$$
$$2$$
$$\frac{1}{2}$$
$$-\frac{1}{2}$$
$$3$$
3. Find $$\frac{dy}{dx}$$ if $$y=\frac{4x-3}{ 2x+1}$$.
$$\frac{10}{(4x-3)^2}$$
$$-\frac{10}{(4x-3)^2}$$
*$$\frac{10}{(2x+1)^2}$$
$$-\frac{10 }{(2x+1)^2}$$
$$2$$
4. Which of the following gives $$\frac{dy}{dx}$$ for $$y = \sin^{4} (3x)$$?
$$4\sin^{3}(3x)\cos(3x)$$
*$$12\sin^{3}(3x)\cos(3x)$$
$$12\sin(3x)\cos(3x)$$
$$12\sin^{3}(3x)$$
$$-12\sin^{3}(3x)\cos(3x)$$
5. Which of the following gives $$y^{\prime\prime}$$ for
$$y=\cos{x}+\tan{x}$$?
*$$-\cos{x} + 2\sec^{2}{x}\tan{x}$$
$$\cos{x}+2\sec^{2}{x}\tan{x}$$
$$-\sin{x}+\sec^{2}{x}$$
$$-\cos{x}+\sec^{2}{x}\tan{x}$$
$$\cos{x}+2\sec^{2}{x}\tan{x}$$
6. Which of the following gives $$\frac{dy}{dx}$$ at $$x=1$$ if $$x^2+2xy=9$$?
$$\frac{11}{2}$$
$$\frac{5}{2}$$
$$\frac{3}{2}$$
$$-\frac{5}{2}$$
*$$-\frac{11}{2}$$
7. Which of the following gives $$\frac{dy}{dx}$$ if $$y=\cos^{3}(3x-2)$$?
*$$-9\cos^{2}(3x-2)\sin(3x-2)$$
$$-3\cos^{2}(3x-2)\sin(3x-2)$$
$$9\cos^{2}(3x-2)\sin(3x-2)$$
$$-9\cos^{2}(3x-2)$$
$$-3\cos^{2}(3x-2)$$
8. Which of the following gives $$\frac{dy}{dx}$$ if $$y=\cos^{-1}(2x)$$?
*$$\frac{- 2}{sqrt(1-4x^2)}$$
$$\frac{- 1}{sqrt(1-4x^2)}$$
$$\frac{2}{sqrt(1-4x^2)}$$
$$\frac{1}{sqrt(1-4x^2)}$$
$$\frac{2x}{sqrt(1-4x^2)}$$
9. Which of the following is true about the graph of $$f(x)=\frac{x^4}{5}$$ at
$$x=0$$?
*It has a cusp
It has a vertical tangent
It has a discontinuity
$$f(0)$$ DNE
none of these
10. Let $$f(x) = \sqrt[3]{x-1}$$. At which of the following points does
$$f^{\prime}(a)\neq \text{ a real number}$$ ?
*$$a=1$$
$$a=-1$$
$$a=2$$
$$a=-2$$
$$a=0$$
11. Let $$y=uv$$ be the product of the functions $$u$$ and $$v$$.
Find $$y^{\prime}(1)$$ if $$u(1)=2$$, $$u^{\prime}(1)=3$$, $$v(1)=-1$$, and
$$v^{\prime}(1)=1$$.
-4
*-1
1
4
7
12. Let $$f(x)=x-\frac{1}{x}$$. Find $$f^{\prime\prime}(x)$$.
$$1+\frac{1}{x^2}$$
$$1-\frac{1}{x^2}$$
$$\frac{2}{x^5}$$
*$$-\frac{2}{x^3}$$
DNE
13. Assume $$f(x)=(x^2-1)(x^2+1)$$. Which of the following gives the number of
horizontal tangents of $$f$$?
0
*1
2
3
4
14. Find the instantaneous rate of change of $$f(x)=\frac{x^2-2}{x+4}$$ at
$$x=-1$$.
$$-\frac{7}{9}$$
$$\frac{5}{9}$$
$$0$$
*$$-\frac{5}{9}$$
$$\frac{7}{9}$$
15. A particle moves along a line so that its position at any time $$t \geq 0$$ is
given by
$$s(t)=2+7t-t^2$$
At which of the following times is the particle moving to the left?
$$t=0$$
$$t=1$$
$$t=2$$
$$t=3$$
*$$t=4$$
16. A particle moves along a line so that its position at any time $$t \geq 0$$ is
given by
$$s(t)=2+7t-t^2$$
At which of the following times is the particle at rest?
$$t=\frac{3}{2}$$
$$t=2$$
*$$t=\frac{7}{2}$$
$$t=4$$
$$t=\frac{5}{2}$$
17. For $$x^2-xy+y^2=1$$ which of the following is equal to $$\frac{dy}{dx}$$?
*$$\frac{y-2x}{2y-x}$$
$$\frac{y+2x}{2y-x}$$
$$\frac{2x}{x-2y}$$
$$\frac{2x+y}{x-2y}$$
$$\frac{y+2x}{x}$$
18. Which of the following is the derivative of $$\tan^{-1}(3x)$$ ?
$$-\frac{3}{1+9x^2}$$
$$\frac{1}{1+9x^2}$$
$$-\frac{1}{1+9x^2}$$
*$$\frac{3}{1+9x^2}$$
$$\frac{3}{\sqrt{1+9x^2}}$$
19. Which of the following gives the slope of the tangent line to the graph of
$$y=e^{2x-4}$$ at $$x=2$$ ?
$$-\frac{1}{2}$$
$$\frac{1}{2}$$
$$-2$$
*$$2$$
$$-\frac{\ln{2}}{2}$$
20. Find $$\frac{dy}{dx}$$ for $$x \cos{y} + y^2 =8$$ .
*$$\frac{\cos{y}}{x \sin{y}-2y}$$
$$\frac{1}{x \sin{y}-2y}$$
$$0$$
$$\frac{- \cos{y}}{x \sin{y}-2y}$$
$$\frac{-1}{x \sin{y}-2y}$$