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The function $b(t)$ gives the value of a particular baseball card in dollars where $t$ is the number of years since the card was purchased. It turns out that $b'(3)=5$. Which statement best explains the meaning of $b'(3)=5$.
Three years after purchase, the card’s value is $5. Over the three years after it was purchased, the card’s value increased by$5 every year. Three years after purchase, the card’s value will increase by $5 over the next year. As time increases by a very small amount from 3 years, the card’s value increases by 5 times as much as the change in time. The instantaneous rate of change of the card’s value with respect to time 3 years after purchase is$5 per year.
The images below show the graph of a logarithmic function given by $y = f(x)$. To the right is an image zoomed in on the point $(4,f(4))$. It turns out that $f'(4)=0.17$. After zooming enough to where the graph of $f$ appears essentially linear, which statement best describes the length of the blue vertical vector?
The length of the blue vector is $4$. The length of the blue vector is $f(4)$. The length of the blue vector is $0.17$. The length of the blue vector is $0.17$ times as much as the value of $4$. The length of the blue vector is $0.17$ times as much as the value of $\Delta x$. The length of the blue vector is $0.17$ times as much as the value of $f(4)$.