{"question":{"question":{"content":"Consider the three expressions and the graph of the function $f$ shown below. **Choose the correct ranking from smallest to largest.** $\qquad \text{I.} \qquad \displaystyle \dfrac{f(4)-f(2)}{2}\ $ $\qquad \text{II.} \qquad \displaystyle \dfrac{f(7)-f(4)}{3}\ $ $\qquad \text{III.} \qquad \displaystyle \lim_{h\to0} \dfrac{f(4+h)-f(4)}{h}\ $ ![](https://ka-perseus-graphie.s3.amazonaws.com/f93185ff0b4833aba4da7f39bdc12c5bd950b59f.png) [[☃ radio 1]]","images":{"https://ka-perseus-graphie.s3.amazonaws.com/f93185ff0b4833aba4da7f39bdc12c5bd950b59f.png":{"width":425,"height":425}},"widgets":{"radio 1":{"type":"radio","alignment":"default","static":false,"graded":true,"options":{"choices":[{"content":"$\text{III} < \text{I} < \text{II}$","correct":true},{"content":"$\text{III} < \text{II} < \text{I}$","correct":false},{"content":"$\text{II} < \text{I} < \text{III}$","correct":false},{"content":"$\text{I} < \text{II} < \text{III}$","correct":false},{"content":"None of these","correct":false}],"randomize":false,"multipleSelect":false,"displayCount":null,"hasNoneOfTheAbove":false,"onePerLine":true,"deselectEnabled":false},"version":{"major":1,"minor":0}}}},"answerArea":{"calculator":false,"chi2Table":false,"periodicTable":false,"tTable":false,"zTable":false},"itemDataVersion":{"major":0,"minor":1},"hints":[{"replace":false,"content":"Of the three expressions, the first two represent slopes of secants and the last represents the slope of the tangent along the curve $f$. The first is the slope of the secant between $x=2$ and $x=4$, the second the slope of the secant between $x=4$ and $x=7$, and the third the slope of the tangent at $x=4.$","images":{},"widgets":{}},{"replace":false,"content":"$\qquad \text{I.} \qquad \displaystyle \dfrac{f(4)-f(2)}{2}\ $ equals the slope of the secant between $x=2$ and $x=4$. We need to imagine this secant and discover that its slope is approximately equal to $\frac{1}{4}$.","images":{},"widgets":{}},{"replace":false,"content":"$\qquad \text{II.} \qquad \displaystyle \dfrac{f(7)-f(4)}{3}\ $ equals the slope of the secant between $x=4$ and $x=7$. We need to imagine this secant and discover that its slope is approximately equal to $\frac{2}{3}$.","images":{},"widgets":{}},{"replace":false,"content":"$\qquad \text{III.} \qquad \displaystyle \lim_{h\to0} \dfrac{f(4+h)-f(4)}{h}\ $ is the slope of the tangent to the curve at $x=4$. From the graph, we can see that the slope is $0$ or very close to $0$.","images":{},"widgets":{}},{"replace":false,"content":"The previous three hints point to one conclusion: $\text{III} < \text{I} < \text{II}$.","images":{},"widgets":{}}]},"problemId":6712,"exerciseId":1013,"exerciseSessionId":null,"hintsEnabled":true,"readOnly":false,"submitterId":null,"checkAnswerButtonText":"Check answer","correctButtonText":"🌟 Yes! You got it!","tryAgainButtonText":"Try again"}

Derivatives basics challenge