NZ Level 8 (NZC) Level 3 (NCEA) [In development]
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Basic derivative of e^x

Interactive practice questions

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

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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=ex is neither strictly positive nor strictly negative.

A

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

B

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

C

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

A

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

B

The gradient function of $y=e^x$y=ex 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=ax2, for $a>0$a>0

A

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

B

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

C

Quadratic, of the form $y'=ax^2$y=ax2, for $a>0$a>0

A

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

B

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

C
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$f\left(x\right)=e^x$f(x)=ex and its tangent line at $x=0$x=0 are graphed on the coordinate axes.

The value of $e^x$ex, where $e$e is the natural base, can be given by the expansion below:

$e^x=1+\frac{x}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\frac{x^5}{5!}+\text{. . .}$ex=1+x1!+x22!+x33!+x44!+x55!+. . .

Outcomes

M8-11

Choose and apply a variety of differentiation, integration, and antidifferentiation techniques to functions and relations, using both analytical and numerical methods

91578

Apply differentiation methods in solving problems

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