MA005: Calculus I

1. Match the graphs of the three functions in Fig. 8 with the graphs of their derivatives.

In problems 3-5, find the slope $\mathrm{m}_{\mathrm{sec}}$ of the secant line through the two given points and then calculate $m_{\tan }=\lim _{h \rightarrow 0} \mathrm{~m}_{\mathrm{sec}}$.

3. $\mathrm{f}(\mathrm{x})=\mathrm{x}^{2}$

(a) $(-2,4),\left(-2+\mathrm{h},(-2+\mathrm{h})^{2}\right)$

(b) $(0.5,0.25),\left(0.5+\mathrm{h},(0.5+\mathrm{h})^{2}\right)$

5. $f(x)=7 x-x^{2}$

(a) $(1,6),\left(1+\mathrm{h}, 7(1+\mathrm{h})-(1+\mathrm{h})^{2}\right)$

(b) $\left(x, 7 x-x^{2}\right),\left(x+h, 7(x+h)-(x+h)^{2}\right)$

7. Use the graph in Fig. 10 to estimate the values of these limits. (It helps to recognize what the limit represents.)

(a) $\lim\limits_{h \rightarrow 0} \frac{f(0+h)-f(0)}{h}$

(b) $\lim\limits_{h \rightarrow 0} \frac{f(1+h)-f(1)}{h}$

(c) $\lim\limits_{h \rightarrow 0} \frac{f(2+h)-1}{h}$

(d) $\lim\limits_{w \rightarrow 0} \frac{f(3+w)-f(3)}{w}$

(e) $\lim\limits_{h \rightarrow 0} \frac{f(4+h)-f(4)}{h}$

(f) $\lim\limits_{s \rightarrow 0} \frac{f(5+s)-f(5)}{s}$

In problems 9 – 11, use the Definition of the derivative to calculate $f '(x)$ and then evaluate $f '(3)$.

9. $f(x) = x^2 + 8$ 

11. $f(x) = 2x^3 – 5x$ 

13. Graph $f(x) = x , g(x) = x^2 + 3$ and $h(x) = x^2 – 5$. Calculate the derivatives of $f$, $g$, and $h$. 

In problems 15 – 17, find the slopes and equations of the lines tangent to $y = f(x)$ at the given points. 

15. $f(x) = x^2 + 8$ at $(1,9)$ and $(–2,12)$ 

17. $f(x) = sin(x)$ at $(π, 0)$ and $(π/2,1)$ 

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

(b) Find the equation of the line perpendicular to the graph of $y = x^2 + 1$ at $(2,5)$. 

(c) Where is the tangent to the graph of $y = x^2 + 1$ horizontal? 

(d) Find the equation of the line tangent to the graph of $y = x^2 + 1$ at the point $(p,q)$. 

(e) Find the point(s) $(p,q)$ on the graph of $y = x^2 + 1$ so the tangent line to the curve at $(p,q)$ goes through the point $(1, –7)$.

21. (a) Find the angle that the tangent line to $y = x^2$ at $(1,1)$ makes with the x–axis. 

(b) Find the angle that the tangent line to $y = x^3$ at $(1,1)$ makes with the x–axis. 

(c) The curves $y = x^2$ and $y = x^3$ intersect at the point $(1,1)$. Find the angle of intersection of the two curves (actually the angle between their tangent lines) at the point $(1,1)$. 

23. Fig. 13 shows the graph of the height of an object at time $t$. Sketch the graph of the object's upward velocity. What are the units for each axis on the velocity graph? 

25. A rock dropped into a deep hole will drop $d(x) = 16x^2$ feet in $x$ seconds. 

(a) How far into the hole will the rock be after 4 seconds? 5 seconds? 

(b) How fast will it be falling at exactly 4 seconds? 5 seconds? $x$ seconds? 

27. It costs $C(x) = √x$ dollars to produce $x$ golf balls. What is the marginal production cost to make a golf ball? What is the marginal production cost when $x = 25$? when $x= 100$? (Include units.) 

29. Define $A(x)$ to be the area bounded by the x–axis, the line $y = x$, and a vertical line at $x$ (Fig. 15). 

(a) Evaluate $A(0), A(1), A(2)$ and $A(3)$. 

(b) Find a formula which represents $A(x)$ for all $x ≥ 0: A(x) =$? 

(c) Determine $\frac{\mathrm{d} \mathrm{A}(\mathrm{x})}{\mathrm{dx}}$.

(d) What does $\frac{\mathrm{d} \mathrm{A}(\mathrm{x})}{\mathrm{dx}}$ represent?

31. Find (a) $D\left(x^{9}\right)$

(b) $\frac{\mathbf{d} x^{2 / 3}}{d x}$

(c) $D\left(\frac{1}{x^{4}}\right)$

(d) $\mathrm{D}\left(\mathrm{x}^{\pi}\right)$

(e) $\frac{\mathbf{d}|\mathrm{x}+5|}{\mathbf{d x}}$

In problems 31 – 37, find a function $f$ which has the given derivative. (Each problem has several correct answers, just find one of them.) 

33. $f^{\prime}(x)=3 x^{2}+8 x$

35. $\frac{\mathbf{d} \mathrm{f}(\mathrm{t})}{\mathrm{dt}}=5 \cos (\mathrm{t})$

37. $\mathrm{D}(\mathrm{f}(\mathrm{x}))=\mathrm{x}+\mathrm{x}^{2}$

Source: Dale Hoffman, https://s3.amazonaws.com/saylordotorg-resources/wwwresources/site/wp-content/uploads/2012/12/MA005-3.2-Definition-of-Derivative.pdf
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