Calculus of Exponential Functions

Consider the graph of $y=e^x$`y`=`e``x`.

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a

Is the function increasing or decreasing?

Increasing

A

Decreasing

B

Increasing

A

Decreasing

B

b

Is the gradient to the curve negative at any point on the curve?

Yes

A

No

B

Yes

A

No

B

c

What does that tell us about the gradient function?

The gradient function of $y=e^x$`y`=`e``x` is neither strictly positive nor strictly negative.

A

The gradient function of $y=e^x$`y`=`e``x` is a strictly positive function.

B

The gradient function of $y=e^x$`y`=`e``x` is a strictly negative function.

C

The gradient function of $y=e^x$`y`=`e``x` is neither strictly positive nor strictly negative.

A

The gradient function of $y=e^x$`y`=`e``x` is a strictly positive function.

B

The gradient function of $y=e^x$`y`=`e``x` is a strictly negative function.

C

d

Which of the following best describes how the gradient is changing on the curve?

As $x$`x` increases, the gradient on the curve is positive but decreasing.

A

As $x$`x` increases, the gradient on the curve is positive and increasing.

B

As $x$`x` increases, the gradient on the curve remains constant.

C

As $x$`x` increases, the gradient on the curve is positive but decreasing.

A

As $x$`x` increases, the gradient on the curve is positive and increasing.

B

As $x$`x` increases, the gradient on the curve remains constant.

C

e

What does that tell us about the gradient function?

$y'$`y`′ must be a function that increases more and more rapidly as $x$`x` increases.

A

$y'$`y`′ must be a function that increases more and more slowly as $x$`x` increases.

B

$y'$`y`′ must be a function that decreases more and more rapidly as $x$`x` increases.

C

$y'$`y`′ must be a function that increases more and more rapidly as $x$`x` increases.

A

$y'$`y`′ must be a function that increases more and more slowly as $x$`x` increases.

B

$y'$`y`′ must be a function that decreases more and more rapidly as $x$`x` increases.

C

f

We have found that the gradient function $y'$`y`′ must be a strictly positive function, and it must also be a function that increases at an increasing rate. What kind of function could $y'$`y`′ be?

Quadratic, of the form $y'=ax^2$`y`′=`a``x`2, for $a>0$`a`>0

A

Exponential, of the form $y'=a^x$`y`′=`a``x`, for $a>0$`a`>0

B

Rational, of the form $y'=\frac{a}{x}$`y`′=`a``x`, for $a>0$`a`>0

C

Quadratic, of the form $y'=ax^2$`y`′=`a``x`2, for $a>0$`a`>0

A

Exponential, of the form $y'=a^x$`y`′=`a``x`, for $a>0$`a`>0

B

Rational, of the form $y'=\frac{a}{x}$`y`′=`a``x`, for $a>0$`a`>0

C

Easy

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