Derivatives & Antiderivatives

Derivatives, Antiderivatives, U-Sub

Hello everyone !  Let's play a game.  For multiple-choice questions, choose the one and only correct answer for each question.  Let's get started!

Multiple Choice

What is the most general antiderivative of 

$$\cos\left(x\right)$$  ?

• $$-\sin\left(x\right)+c$$  

• $$\sin\left(x\right)$$  

• $$\sin\left(x\right)+c$$  

• $$-\sin\left(x\right)$$  

Multiple Choice

What is the derivative of   $$\tan\left(x\right)$$  ?

• $$\cot\left(x\right)$$  

• $$\sec\left(x\right)\tan\left(x\right)$$  

• $$\sec^2\left(x\right)$$  

• $$-\csc\left(x\right)\cot\left(x\right)$$  

Multiple Choice

What is the most general antiderivative of  $$f\left(x\right)=\frac{1}{x}\ \$$ on the interval  $$\left(-\infty,\ 0\right)\ ?$$  

• $$-\frac{1}{x^2}+c$$  

• $$\ln\left(x\right)+c$$  

• $$x^{-2}+c$$  

• $$\ln\left|x\right|+c$$  

Multiple Choice

Differentiate 

$$H\left(x\right)=\left(x-\sqrt{x}\right)\left(x+\sqrt{x}\right)$$  

• $$2x-1$$  

• $$2x+1$$  

• $$\left(1-\frac{1}{2\sqrt{x}}\right)\left(1+\frac{1}{2\sqrt{x}}\right)$$  

• $$x^2-x$$  

Multiple Choice

Differentiate 

$$y=\sec\left(\theta\right)\tan\left(\theta\right)$$  

• $$\sec\left(\theta\right)$$  

• $$\sec^3\left(\theta\right)\tan\left(\theta\right)$$  

• $$\sec\left(\theta\right)\left(\tan^2\theta+\sec^2\theta\right)$$  

• $$\tan\left(\theta\right)\left(\tan^2\theta+\sec^2\theta\right)$$  

Multiple Choice

$$y=\frac{\sqrt{x}}{8+x}$$  

Differentiate

• $$\frac{8-x}{2\sqrt{x}\left(x+8\right)^2}$$  

• $$\frac{8+x}{2\sqrt{x}\left(x+8\right)^2}$$  

• $$\frac{8-x}{2x\left(x+8\right)^2}$$  

• $$\frac{8-x}{2\sqrt{x}\left(x+8\right)^{ }}$$  

Multiple Choice

In the videos of our Lessons,

U-sub was illustrated as....

• an U-shaped sandwich from Subway.

• an U-shaped subway in New York.

• a submarine named U-Sub.

Multiple Choice

 Find $$\int_{ }^{ }x^3\cos\left(x^4+2\right)dx$$  

• $$\frac{1}{2}\cos\left(x^4+2\right)+c$$  

• $$\frac{1}{3}\cos\left(x^4+2\right)+c$$  

• $$-\frac{1}{4}\sin\left(x^4+2\right)+c$$  

• $$\frac{1}{4}\sin\left(x^4+2\right)+c$$  

Multiple Choice

Integration and differentiation are inverse processes.  In the videos of our lessons, we learned that the substitution rule (U-Sub) in integration corresponds to  __________ in differentiation.

• the Product rule

• the Quotient rule

• the Power rule

• the Chain  rule

Multiple Choice

How should I check if my integration result is correct?

• Google it.

• Ask my friends.

• Try differentiate the result and see if I get back an answer the same as the integrand.

Open Ended

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