Limits. Final Exam Study Guide. Calculus I. 1. Basic Limits I: Evaluate each limit exactly. (a) lim. (c) lim. 2t 15 3 (g) lim. (e) lim. (f) lim.

Transcription

1 Limits 1. Basic Limits I: Evaluate each limit eactly. 3 ( +5 8) (c) lim(sin(α) 5cos(α)) α π 6 (e) lim t t 15 3 (g) lim t 0 t (4t 3 8t +1) t 1 (tan(θ) cot(θ)+1) θ π 4 (f) lim 16 ( 5 (h) lim t 0 3 t ). Basic Techniques I: Evaluate each limit, given that lim f() = 6, lim h() = 3 and lim g() = 11. (f() 3g()) (c) lim(f()h()) h() (e) lim f()+g() (f() 1) f()+g() h()+4 (f) lim h()+3 f() g() 3. Basic Limits II: Evaluate each limit eactly. t +5t 6 t 1 [ t 1][ ] 1 (c) lim sin(θ) θ 0 θ 4. Intermediate Limits I: Evaluate each limit eactly (c) lim sin() 5 (e) lim θ 0 tan(θ) θ sin(θ π/4) (g) lim θ π/4 θ π/4 (Hint: sin() = sin() cos()) (Hint: t = θ π 4 ) z 11 z 11 z 11 sin() 5 (f) lim z 0 (h) lim t 0 3z sin(z) ( t t t t (Hint: t = ) ) Created/Revised 11/7/18 1

2 5. Eamples I: Provide an appropriate eample, if possible. Each part is separate problem. If no such eample eists, then write impossible in the answer blank. Give an eample of a function: (a) f and a c R such that lim f() = 0. (b) g such that lim 3 g() = D.N.E., but lim g() =. (c) h such that lim h() = 5 and lim 3h() = 10. (d) f and a c R such that lim f() = 0, but lim (f()) = Intermediate Limits II: Evaluate each limit eactly. t 0 3tan(t) 11t (c) lim t 3t 1 t sin () Advanced Limits I: Evaluate each limit eactly. t+ t 8 3t 4 (c) lim 1 cos() sin() (e) lim sin() 1 cos() sin() ( ) (f) lim sin 8. Eamples II: Provide an appropriate eample, if possible. Each part is separate problem. If no such eample eists, then write impossible in the answer blank. Give an eample of a function: (a) f and a function g such that lim f() = 0, lim g() D.N.E., but lim (f()g()) eists. f() (b) f and a function g such that lim f() = 0, lim g() = 0, but lim g() eists. f() (c) f and a function g such that lim f() = 0, lim g() = 0, but lim g() D.N.E. Created/Revised 11/7/18

3 9. Intermediate Limits III - One Sided Limits: Evaluate each limit eactly. t t 0 + t (c) lim t + 3t 7 t The Derivative sin() 1 1 Eample: Use the definition of the derivative to differentiate f() = f f(+h) f() () lim h 0 h 19(+h) = lim h 0 h 19+19h = lim h 0 h 19+19h = lim h 0 h ( 19+19h = lim h ) h 0 h h h+11 (19+11) = lim h 0 h ( 19+19h ) 19h = lim h 0 h ( 19+19h ) 19 = lim h h = = Definition of the Derivative Use the definition of the derivative to find f () for (a) f() = ++1 (b) f() = sin() Created/Revised 11/7/18 3

4 11. Basic Differentiation: Find the derivative of each function. (a) m() = (b) s() = (c) g(θ) = 6sin(θ)+4cos(θ)+8θ (d) h(t) = 6sec(t)+7cot(t) (e) p() = (f) r(α) = 3csc(α) tan(α) 3 1. Slope of the Tangent Line: Find the slope of the tangent line to: (a) f() = at (1,). (b) g() = sin()+1 at ( π,3). (c) h() = +3 at (4,). (d) m() = 1 + at (1,3). 13. Equation of the Tangent Line: Find the equation of the tangent line to: (a) f() = at (1,). (b) g() = sin()+1 at ( π,3). (c) h() = +3 at (4,). (d) m() = 1 + at (1,3). 14. Eamples III: Provide an eample, if possible, of: (a) a function that is continuous at = 3, but not differentiable at = 3. (b) a function that is differentiable at = 3, but not continuous at = 3. (c) a function that has a vertical tangent line only at = 3. (d) a function that has infinitely many points at which the tangent line is horizontal. (e) a continuous function that has infinitely many points at which the tangent line is vertical. 15. Basic Product Rule: Use the product rule to find the derivative of each function. (a) f(θ) = sin(θ)cos(θ) (b) g(t) = tsin(t) (c) h() = ( 3)( 5 )(7+3) (d) m() = 1 sin() 16. Basic Quotient Rule: Use the quotient rule to find the derivative of each function. (a) f() = (b) g(t) = tsin(t) 6+8 (c) h() = +5 (d) m() = sin() Basic Chain Rule: Use the chain rule to find the derivative of each function. (a) f() = 5(8 6) 5 (b) g(t) = 5sin (t) (c) f() = 1 7 ( 6+8) (d) m() = 4sin(7)+8 Created/Revised 11/7/18 4

5 18. Basic Chain Rule: Usethechain rule toestablish a derivative formula forf() =. 19. Intermediate Derivatives I - A Miture of Rules Use the appropriate combination of derivative rules to find the derivative of each function. t+1 a) f(t) = b) f() = 5 sin(t) cos(t) sin(1 3) c) g(z) = ( z 3 6z 1 ) 3( z +4z 6 ) ( ) d) m() = Intermediate Derivatives II - A Miture of Rules Use the appropriate combination of derivateve rules to find the derivative of each function. (t+1) a) f(t) = b) f() = sin(5 ) sin(t) cos(3t) + 1 ( 11 c) g(z) = 5+4 ) 3 ( Implicit Differentiation I: Find dy d ) 5 d) m() = (5 1)5 for each: (17 4) 4 a) +y = 1. b) sin(y) =. c) 10y y 3 = +3y. d) y 1 = y 1 5y 4.. Implicit Differentiation II: Find the slope of the tangent line to the graph at the indicated point. ( 3 ) a) +y = 1; at. b) sin(y) = ; at ( ) 1, π c) 10y y 3 = +3y; at (1, ) y 1 d) y 1 = ; at (1, 1). 5y 4 3. Related Rates I: Find the indicated (instantaneous) rate of change. a) A water tank has the shape of an inverted circular cone with base radius m and height 4 m. Water is being pumped into the tank at the rate of m3 min. Find the rate at which the water is rising at the instant when the water is 3m deep. b) A spherical balloon is being deflated so that its radius decreases at cm min. Find the rate at which the volume is decreasing at the instant when the radius is 4cm. c) Gravel is being dumped from a conveyor belt at a rate of 30 ft3 min and its coarseness is such that it forms a pile in the shape of a right circular cone whose base diameter and height are always equal. Find the rate at which the height of the conical pile is increasing at the instant when the it is 10 ft high? Created/Revised 11/7/18 5

6 Applications of the Derivative 4. Analyze Function: Let f() = Answer each of the following: (a) State the domain (c) State any horizonal/vertical asymptotes (e) Find f () (g) Find any points of horizontal tangency Identify the points at which f has a (i) relative maimum or a (k) Identify the possible inflection point(s) (m) Identify the inflection point(s) (b) Find the - and y-intercepts (d) Find f () (f) Identify the critical point(s) (h) Complete a sign chart for f () (j) relative minimum (l) Complete a sign chart for f () 5. Analyze Function: Let g() = ( 3). Answer each of the following: 7 (a) State the domain (b) Find the - and y-intercepts (c) State any horizonal/vertical asymptotes (e) Find g () (g) Find any points of horizontal tangency Identify the points at which g has a (i) relative maimum or a (k) Identify the possible inflection point(s) (m) Identify the inflection point(s) (d) Find g () (f) Identify the critical point(s) (h) Complete a sign chart for g () (j) relative minimum (l) Complete a sign chart for g () 6. Analyze Function: Let h() = Answer each of the following: (a) State the domain (c) State any horizonal/vertical asymptotes (e) Find h () (g) Find any points of horizontal tangency Identify the points at which h has a (i) relative maimum or a (k) Identify the possible inflection point(s) (m) Identify the inflection point(s) (b) Find the - and y-intercepts (d) Find h () (f) Identify the critical point(s) (h) Complete a sign chart for h () (j) relative minimum (l) Complete a sign chart for h () Created/Revised 11/7/18 6

7 7. Analyze Function: Let t() = sin() on (0,π). Answer each of the following: (a) State the domain (c) State any horizonal/vertical asymptotes (e) Find t () (g) Find any points of horizontal tangency Identify the points at which t has a (i) relative maimum or a (k) Identify the possible inflection point(s) (m) Identify the inflection point(s) (b) Find the - and y-intercepts (d) Find t () (f) Identify the critical point(s) (h) Complete a sign chart for t () (j) relative minimum (l) Complete a sign chart for t () 8. Optimization (a) Show that of all the rectangles with a fied perimeter k the square has the largest area. (b) Find two positive numbers subject to the condition that the sum of the first and twice the second is 00 and the product is maimal. (c) A rancher has 400 feet of fencing with which to enclose two adjacent rectangular corrals. What dimensions should be used so that the enclosed area is maimal? (d) Find two positive numbers subject to the condition that their product is 384 and the sum of the first pluse thrice the second is minimal. (e) Find the dimensions of the largest (in terms of area) rectangle that can be inscribed in a semicircle of radius r. 9. Limits at Infinity: Evaluate each limit, if it eists sin() (c) lim (e) lim 1 (f) lim (g) lim 5+ (h) lim 16+ (i) lim( ) (k) limcos ( ) 1 (Hint: Think Continuity) (j) lim ( 7+4 +) (l) lim sin ( 1 ) (Hint: Let t = 1 ) Created/Revised 11/7/18 7

8 30. Absolute Etrema: Find all etrema of the given function on the indicated interval. (a) f() = +3 on [0,3] [ (c) p() = cos(π) on 0, 1 ] 6 (b) g() = +1 on [ 1,1] (d) g() = 5( 3) on [ 4, 1] 31. Understanding the Concepts: True or False T F The maimum value of a continuous function on a closed interval may occur at two different values in the interval. T F If a f is continuous on [a,b], then f must have a maimum on [a,b]. T F If a f is continuous on (a,b), then f must have a minimum on (a,b). T F If f has a critical point at c, then g() = f()+8 also has a critical point at c. T F The Mean Value Theorem may be applied to f() = 1 T F The Mean Value Theorem may be applied to f() = 1 on the interval [ 1,]. on the interval [1,]. T F If a function f has three distinct intercepts, then f must have at least two critical points at which the tangent line is horizontal. T F If a polynomial function f has three distinct intercepts, then f must have at least two critical points at which the tangent line is horizontal. T F If f is increasing on its domain, then f is decreasing on its domain. T F f() = 1 T F f() = 1 is concave down when < 0 and concave up when > 0. has an inflection point at 0. T F If f (c) = 0 and f (c) = 3, then f has a relative maimum at c. T F If f(1) = 0 and f() = 0, then Rolle s Theorem may be applied to f on [1, ]. T F If f(1) = 0 and f() = 0 and f is continuous on [0, ], then Rolle s Theorem may be applied to f on [1, ]. T F If f(1) = 0 and f() = 0 and f is differentiable on (0, ), then Rolle s Theorem may be applied to f on [1, ]. T F If f(1) = 0,f() = 0, and f is differentiable on (, ), then Rolle s Theorem may be applied to f on [1, ]. Created/Revised 11/7/18 8

9 Integration 3. Basic Integration - Part I: Evaluate each indefinite integral. (a) cos(θ)dθ ) (t t+4)dt ( 1 (b) )d (c) (sec(α) tan(α) sin(α)) dα (csc (d) (θ) csc(θ)cot(θ) ) dθ (e) πdθ 33. Basic Integration - Part II: Evaluate each indefinite integral. 3 (a) d (b) t (t 3 t 3 +4)dt ( 3 (c) u u + 5u ) 1 5 u +8 du (d) 3 d (3cos (e) (θ)+3sin (θ) ) dθ (g) 0dz 34. General Solution to dy d = f(): Find the general solution of the differential equation. (a) dy d = 3 5 (c) dy dθ = θ θ (b) dy dt = 5cos(t) 3sin(t) (d) dy 5 dy = Intermediate Integration - Part I - - Substitution: Evaluate the indefinite integral using the change-of-variables technique. 3 d (a) (b) sin(3θ)dθ (c) (3+1) d (d) ( ) d 36. Understanding the Concepts: Verify the given integration formula. Verify tsin(t)dt = tcos(t)+ 1 sin(t). Created/Revised 11/7/18 9

10 37. Intermediate Integration - Part II - The Definite Integral Evaluate each indefinite integral using the Fundamental Theorem of Calculus. (a) (c) (e) π π 4 0 π 4 π 3 cos(θ)dθ (b) 3 1 (t t+4)dt ( π 1 4 )d (d) (sec(α) tan(α) sin(α)) dα π 6 ( csc (θ) csc(θ)cot(θ) ) dθ (f) 5 πdθ 38. Advanced Integration - Part I - The Definite Integral & Substitution Evaluate each indefinite integral using the Fundamental Theorem of Calculus and the change-of-variables technique. (a) (c) d (3+1) d π (b) sin(3θ)dθ (d) π ( ) d 39. Advanced Integration - Part II Use the Fundamental Theorem of Calculus to find the derivative of each function. π (a) f() = 5t+4dt (b) g(t) = sin(5) d (c) h() = dt (d) m(θ) = (1+7t) t sin(θ) cos(θ) d Created/Revised 11/7/18 10

11 f() = ln (( + 1) 3 ) f () f () f() = ln ( ) s() = ln () g() = (ln()) t() = e h() = 7e sin() f(t) = e t + ln(5t 7) h() = 3 ln sin() f(t) = 4 ln + 5 m() = 1 π e / f() = e tan() sec() ( (3 f + 4) 3 ) () f() = ln 3 5 dy d y = 1 f() = e ln > 0 f () ln( + 5) = ln() + ln(5) f() = ln(π) f () = 1 π (ln ) = ln() ln() = 1 1 = ln c c > 0 f f f() = n n f e a b = e a b f() = ln() f(e n+1 ) f(e n ) = 1 n 1

12 d (5e + 6)d ( 5 tan(θ) )dθ 3 d sec(θ)dθ d (5e + 6)d d π/4 π/4 sec(θ)dθ (5π cot(θ) + 3 csc(θ))dθ π/4 0 π/ π/4 (5 tan(θ) )dθ (5π cot(θ) + 3 csc(θ))dθ dt 1 t d ln() d θ sec(θ )dθ 4 cos(t) e sin(t) dt 1 tan(ln())d e e d 4 e d 0 sin() 5 cos() 47 d cot() ln(sin())d e 4 d 4 + e4 1