Q: A population of bacteria grows according to $$ P(t) = 100e$$^{0.03t}$$ . $$What is the instantaneous rate of growth at $$ t = 10 $$? (Use the chain rule.)

$$ 100e$$^{0.3}$$ \\times 0.03 $$

Question 1

Given the graph of a function with a sharp corner at the point $$ (1, 2) $$, what is the value of the derivative at that point?

Accepted Answer

Undefined

Question 2

Estimate the slope of the curve at the indicated point $$ (0, 0) $$ for a function that appears to be decreasing at a $$ 45^\circ $$ angle.

Accepted Answer

$$ -1 $$

Question 3

Estimate the slope of the curve at the indicated point $$ (-1, 0) $$ for a function that is horizontal at that point.

Accepted Answer

$$ 0 $$

Question 4

Using the definition of the derivative, what is $$ f'(x) $$ for $$ f(x) = 5x^2$ $$?

Accepted Answer

$$ 10x $$

Question 5

Given $$ f(x) = x^2$ - 9 $$, what is $$ f'(x) $$?

Accepted Answer

$$ 2x $$

Question 6

If $$ f(x) = x^2$ - 9 $$, what is $$ f'(0) $$?

Accepted Answer

$$ 0 $$

Question 7

If $$ f(x) = x^2$ - 9 $$, what is $$ f'(3) $$?

Accepted Answer

$$ 6 $$

Question 8

Find the equation of the tangent line to $$ y = x^2$ - 4  at $$the point $$ (-2, 0) .$$

Accepted Answer

$$ y = 4x - 8 $$

Question 9

Find the equation of the tangent line to $$ y = x^2$ + 1  at $$the point $$ (-2, 5) .$$

Accepted Answer

$$ y = -4x - 3 $$

Question 10

Suppose the position of a car is given by $$ s(t) = t^2$ - 4t + 3 . $$What is the instantaneous velocity at $$ t = 2 $$? (Recall that velocity is the derivative of position.)

Accepted Answer

$$ 0 $$

Question 11

A chemical reaction's concentration is modeled by $$ C(t) = 5e$$^{-2t}$$ . $$What is the rate of change of concentration at $$ t = 0 $$? (Use the chain rule.)

Accepted Answer

$$ -10 $$

Question 12

If $$ f(x) = x^3$ - 3x^2$ + 2x $$, at which $$ x -$$value does the tangent line have a slope of $$ 0 $$? (Find all real solutions.)

Accepted Answer

Question 13

A population of bacteria grows according to $$ P(t) = 100e$$^{0.03t}$$ . $$What is the instantaneous rate of growth at $$ t = 10 $$? (Use the chain rule.)

Accepted Answer

$$ 100e$$^{0.3}$$ \times 0.03 $$

Question 14

Which of the following best describes the meaning of the derivative at a point $$ x = a $$?

Accepted Answer

The slope of the tangent line to the curve at $$ x = a $$

Question 15

If $$ f(x)  is $$not differentiable at $$ x = a $$, which of the following could be true?

Accepted Answer

The graph of $$ f(x) $$ has a sharp corner at $$ x = a $$

Question 16

A ball is thrown upward and its height is given by $$ h(t) = -16t^2$ + 32t + 5 . $$What is the velocity of the ball at $$ t = 1 $$ second?

Accepted Answer

$$ 16 $$

Question 17

If $$ f(x) = |x| $$, what is $$ f'(0) $$?

Accepted Answer

Undefined

Question 18

Given $$ f(x) = x^2$ - 4x + 7 $$, what is the equation of the tangent line at $$ x = 1 $$?

Accepted Answer

$$ y = -2x + 6 $$

Question 19

A medication's concentration in the bloodstream is modeled by $$ C(t) = 50e$$^{-0.1t}$$ . $$What is the rate of change of concentration at $$ t = 5 $$?

Accepted Answer

$$ -5e$$^{-0.5}$$ $$

Derivatives and Tangent Lines: Calculus Study Notes

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