Finding the First Derivative

Derivative Review Day 1

Multiple Choice

$$\lim_{\theta\rightarrow0}\ \frac{3\sin\left(3\theta\right)}{3\theta}$$  

• 0

• 1

• 3

• The Limit Does Not Exist

Multiple Choice

$$\lim_{x\rightarrow5}\ \frac{x^2-3x-10}{x^2-x-20}$$  

• $$0$$  

• $$\frac{7}{9}$$  

• $$1$$  

• The Limit Does Not Exist

Multiple Choice

Find $$\frac{dy}{dx}$$ if  $$y=\left(\sin x\right)\left(\ln x\right)$$  

• $$\frac{\cos x}{x}$$  

• $$\frac{\sin x}{x}+\cos x\ln x$$  

• $$\frac{\sin x}{x}$$  

• $$\frac{\sin x}{x}-\cos x\ln x$$  

Multiple Choice

Find $$\frac{dy}{dx}$$ if  $$y=\sin^3\left(x^2\right)$$  

• $$6x\sin^2\left(x^2\right)\cos\left(x^2\right)$$  

• $$3\cos^2\left(x^2\right)$$  

• $$\cos^3\left(2x\right)$$  

• $$6x\sin\left(x^2\right)\cos\left(x^2\right)$$  

Multiple Choice

Find $$\frac{d^2y}{dx^2}$$ if  $$y=xe^{-x}$$  

• $$-e^{-x}$$  

• $$e^{-x}\left(x+2\right)$$  

• $$e^{-x}\left(x-2\right)$$  

• $$e^{-x}$$  

Multiple Choice

Find $$\frac{dy}{dx}$$  if $$y=\frac{x^6-4x^4+3x^3}{x^2}$$  

• $$4x^3-8x+3$$  

• $$\frac{6x^5-16x^3+9x^2}{2x}$$  

• $$3x^2-8x+\frac{9}{2}x^{\frac{1}{2}}$$  

• $$4x^4-16x^3+9x^2$$  

Multiple Choice

Find $$\frac{dy}{dx}$$  if $$y=\frac{\sec^2\left(x\right)-\tan^2\left(x\right)}{\csc\left(x\right)}$$

• $$-\csc x\cot x$$  

• $$\cos x$$  

• $$2\sin x\cos x$$  

• $$\sin x$$  

Multiple Choice

The Intermediate Value Theorem guarantees a value  $$c$$  such that  $$f\left(c\right)=0$$  for  $$f\left(x\right)=2x^2+x-4$$  on which of the following intervals?

• (-1,0)

• (0,1)

• (1,2)

• (2,3)

Multiple Choice

What value of $$k$$ makes  $$f\left(x\right)$$  continuous for all values of x?

We get  $$f\left(x\right)=3x^2+5kx-44$$   for  $$x\ge2$$  

and  $$f\left(x\right)=2kx-x^3$$  for  $$x<2$$  

• $$-4$$  

• $$\frac{12}{7}$$  

• $$4$$  

• $$\frac{20}{3}$$  

Multiple Choice

Find $$\frac{dy}{dx}$$  if $$y=\frac{\sin\left(e^x\right)}{1+\cos\left(e^x\right)}$$  

• $$\frac{e^x}{1+\cos\left(e^x\right)}$$  

• $$\frac{e^x}{1-\cos\left(e^x\right)}$$  

• $$\frac{e^x}{\left(1+\cos\left(e^x\right)\right)^2}$$  

• $$\frac{e^x}{\left(1-\cos\left(e^x\right)\right)^2}$$