Question 1

Estimate the slope of the curve at point P in the first image, where the tangent line is drawn.

Accepted Answer

$$-2$$

Question 2

Differentiate $$f(x) = x^3$.

Accepted Answer

$$3x^2$$

Question 3

Find the derivative of $$f(x) = \sqrt{x^2$ - 10x + 37}.$$

Accepted Answer

$$\frac{2x-10}{2\sqrt{x^2$-10x+37}}$$

Question 4

Differentiate $$y = 4x^2$(x-5)^3$.

Accepted Answer

$$8x(x-5)^3$$ + $$12x^2(x-5)^2$

Question 5

Differentiate $$f(x) = 5e$$^{2x}$$.

Accepted Answer

$$10e^{2x}$$

Question 6

Differentiate $$f(x) = \ln(7x^2$ - e$$^{3x}$$).$$

Accepted Answer

$$\frac{14x - 3e$$^{3x}$$}{7x^2$ - e$$^{3x}$$}$$

Question 7

Differentiate $$y = \frac{2x}{x^2$ + 1}.$$

Accepted Answer

$$\frac{2(x^2$+1) - 4x^2$}{(x^2$+1)^2$$}$

Question 8

Write the equation of the tangent line to the curve $$y = x^2$ + 4x - 2 at$$ $$x = 1.$$

Accepted Answer

$$y = 6x - 7$$

Question 9

Given $$y = (x^2$ - 24)^3$, find the equation of the tangent line at x = 5$.

Accepted Answer

$$y = 27x - 337$$

Question 10

Compute $$v''(2)$$, where $$v(t) = 4t^3$ + 2t.$$

Accepted Answer

$$48$$

Question 11

If $$y = (x^2$ + 7)^5$, find $$\frac{$$d^2y$$}{$$dx^2$$}$$.

Accepted Answer

Question 12

For the function $$f(x, y) = 9y^2$e^{x}$$, find $$\frac{\partial f}{\partial x}$$ and $$\frac{\partial f}{\partial y}$.

Accepted Answer

$$\frac{\partial f}{\partial x} = 9y^2$e^{x}$$, \ \frac{\partial f}{\partial y} = $$18ye^{x}$

Question 13

For the function $$f(x, y) = 4x^2$ + 5xy - 9y^2$ + 5x - 7y - 3$$, find $$\frac{\partial f}{\partial x}$$ and $$\frac{\partial f}{\partial y}.$$

Accepted Answer

$$\frac{\partial f}{\partial x} = 8x + 5y + 5, \ \frac{\partial f}{\partial y} = 5x - 18y - 7$$

Question 14

For the function $$f(x, y) = \frac{2x}{x^2$ + 3y}$$, find $$\frac{\partial f}{\partial x}$$ and $$\frac{\partial f}{\partial y}.$$

Accepted Answer

$$\frac{\partial f}{\partial x} = \frac{2(x^2$+3y) - 4x^2$}{(x^2$+3y)^2$$}, \ \frac{\partial f}{\partial y} = \frac{-6x}{$$(x^2+3y)^2$}$$

Question 15

Find and simplify the difference quotient for $$f(x) = 7x^2$ - 3x.$$

Accepted Answer

$$14x + 7h - 3$$

Question 16

Using the second image, for which $$x$$ values is $$f'(x) = 0$$?

Accepted Answer

$$x \approx 3, 8, 13$$

Question 17

A company’s profit function is $$P(x) = -2x^2$ + 40x - 100$$, where $$x is $$the number of units sold. At what value of $$x is $$the profit maximized?

Accepted Answer

$$x = 10$$

Question 18

Suppose the cost to produce $$x$$ items is $$C(x) = 100 + 5x + 0.1x^2$. What is the marginal cost when x = 20$?

Accepted Answer

$$9$$

Business Calculus Exam 1 Review – Step-by-Step Guidance

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